1) The number of pieces that April has is: 9 pieces
2) The length of each piece is: 4 inches
How to solve Ratio and Proportion Problems?A ratio is defined as an ordered couple of numbers a and b, written as a/b where b can not equal 0. A proportion is defined as an equation in which two ratios are set equal to each other.
April has a sheet of paper that is 3 feet long.
She cuts the length of paper into thirds and then cuts the length of each of these 1/3 pieces into sixths.
A paper is divided into 3 equal parts then the number of pieces is 3.
She again divided the 3rd part of the paper into 3 equal parts. Then the number of equal parts of the 3rd part paper is 3.
a. Then the total pieces will be
Total pieces = 3 × 3
Total pieces = 9
The total pieces is 9
b. The length of each part will be;
3/9 = 1/3 ft
Thus:
Length of each = 1/3 * 12 inches = 4 inches
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If 15. 0 g of CaCl2 are present in 250 mL of aqueous solution, what is the concentration of CaCl2 in % (w/v)?
The concentration of CaCl2 in % (w/v) can be calculated by dividing the mass of CaCl2 by the volume of the solution and multiplying by 100. In this case, if 15.0 g of CaCl2 is present in 250 mL of solution, the concentration of CaCl2 can be determined.
The concentration of a solution is commonly expressed as a percentage by weight/volume (% w/v), which represents the mass of solute (in this case, CaCl2) in a given volume of solution. To calculate the concentration, divide the mass of CaCl2 (15.0 g) by the volume of the solution (250 mL) and multiply by 100.
Concentration of CaCl2 in % (w/v) = (Mass of CaCl2 / Volume of Solution) x 100
Substituting the given values:
Concentration of CaCl2 in % (w/v) = (15.0 g / 250 mL) x 100
Simplifying the expression:
Concentration of CaCl2 in % (w/v) = 6.0% (rounded to one decimal place)
Therefore, the concentration of CaCl2 in the given solution is 6.0% (w/v).
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A shop sells two brands of eggs. Brand A and Brand B , Their prices are given in the ratio 4:7, if a baker buys 8 trays of brand A and 6 trays of Brand B at a total cost of R279,45
How much did the baker pay for the 8 trays of brand
The baker paid approximately R15.08 for each tray of Brand A.
Understanding Word ProblemLet:
x = price of each tray of Brand A
y = price of each tray of Brand B
Given that the prices are in the ratio 4:7, we can write the equation:
x/y = 4/7
To find the individual prices of Brand A and Brand B, we can introduce a constant k:
x = 4k
y = 7k
The total cost of 8 trays of Brand A (8x) and 6 trays of Brand B (6y) is R279.45:
8x + 6y = 279.45
Substituting the expressions for x and y:
8(4k) + 6(7k) = 279.45
32k + 42k = 279.45
74k = 279.45
Dividing both sides by 74:
k = 279.45 / 74
k ≈ 3.77
Now we can find the price of each tray of Brand A:
x = 4k ≈ 4 * 3.77 ≈ R15.08
Therefore, the baker paid approximately R15.08 for each tray of Brand A.
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Maggie has $14,100 to invest, and wishes to gain $4,000 in interest over the next eight years. Approximately what is the minimum simple interest rate Maggie needs to reach her goal? a. 2. 89% b. 3. 55% c. 4. 95% d. 5. 18% Please select the best answer from the choices provided A B C D.
Maggie needs to earn $4,000 in interest over eight years on her $14,100 investment. Rounded to two decimal places, the minimum simple interest rate Maggie needs is approximately 5.18%.
To find the minimum simple interest rate Maggie needs, we can use the formula for simple interest:
Interest = (Principal × Rate × Time)
Here, Maggie's goal is to earn $4,000 in interest over eight years, and she has an initial investment of $14,100.
Let's assume the simple interest rate Maggie needs is "r."
Using the formula, we can write the equation as:
$4,000 = ($14,100 × r × 8)
To solve for "r," we divide both sides of the equation by ($14,100 × 8):
r = $4,000 / ($14,100 × 8)
Calculating this gives us:
r ≈ 0.0358
To convert this to a percentage, we multiply by 100:
r ≈ 3.58%
Rounded to two decimal places, the minimum simple interest rate Maggie needs is approximately 3.58%. However, the provided answer choices are given with different rounding. Among the options provided, the closest value to 3.58% is d. 5.18%.
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If Firm A and Firm B both produce a laptop at the same total cost, but the reservation price for Firm A's laptop is $1,000 and the reservation price for Firm B's laptop is $1,200, who has the competitive advantage
In this scenario, Firm B has the competitive advantage over Firm A. The reservation price represents the maximum price that a consumer is willing to pay for a product.
A higher reservation price indicates that consumers perceive more value in the product and are willing to pay a higher price for it.
Since Firm B's laptop has a reservation price of $1,200, compared to Firm A's reservation price of $1,000, it implies that consumers are willing to pay more for Firm B's laptop. This indicates that Firm B's laptop is perceived as having a higher value or quality by consumers.
Given that both firms produce the laptop at the same total cost, Firm B's higher reservation price indicates that it can potentially charge a higher price for its laptop while still remaining competitive in the market. This gives Firm B a competitive advantage over Firm A, as it has the opportunity to generate higher profits per unit sold or capture a larger market share by offering a product with a higher perceived value.
It's worth noting that other factors, such as marketing strategies, product features, brand reputation, and customer preferences, can also influence the competitive advantage between the two firms. However, based on the given information regarding the reservation prices, Firm B appears to have the competitive advantage in terms of consumer willingness to pay a higher price for its laptop.
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A grocer mixes together some cashews costing $8 per kilogram with some Brazil nuts costing
$10 per kilogram. The grocer sold 12 kg if the mixture for $8.50 per kilogram. How many
kilograms of cashews were in the mixture the grocer sold?
I know the answer is 9 but how do i get that?
9 kilograms of cashews were in the mixture the grocer sold.
To solve the given problem, let x represent the number of kilograms of cashews.
Hence, the number of kilograms of Brazil nuts would be (12 - x) as the grocer sold 12 kg of the mixture.
Therefore, the cost of the cashews at $8 per kilogram is 8(x)
and the cost of the Brazil nuts at $10 per kilogram is 10(12 - x).
Hence, the cost of the mixture at $8.50 per kilogram is:
8.5*(12) = 8(x) + 10(12 - x)
We solve this equation for x:
102 = 8(x) + 120 - 10(x)
2(x)= 18
x = 9
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Let â D be an acute angle such that tanD=0. 28. Use a calculator to approximate the measure of â D to the nearest tenth of a degree. What is the measurement of Please show all the work on how you got your answer.
Given that tan D = 0.28 To approximate the value of D, we can use the inverse tangent function tan⁻¹(0.28) on a calculator:
D ≈ 15.9° (rounded to one decimal place)
Therefore, the measurement of angle D to the nearest tenth of a degree is approximately 15.9°.Explanation:We know that tangent of angle D is 0.28.tan D = 0.28 To find the value of D, we need to take the inverse tangent of 0.28.
i.e, D = tan⁻¹(0.28)We use a calculator to evaluate this expression.
D ≈ 15.9° (rounded to one decimal place)
Therefore, the measurement of angle D to the nearest tenth of a degree is approximately 15.9°.
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A cylinder and a cone have the same height and the same base areas. If the volume of the cylinder is 66 cubic inches, what is the volume of the cone? (Use 3. 14 for Pi)
Answer:
The volume of the cone is 22 cubic inches.
Step-by-step explanation:
To find the volume of the cone, we need to use the formula for the volume of a cone:
Volume of a cone = (1/3) * π * r^2 * h
Given that the height and base area of the cylinder and cone are the same, we can assume that the radius of the cylinder's base is equal to the radius of the cone's base.
We know that the volume of the cylinder is 66 cubic inches, so we can set up the equation:
66 = π * r^2 * h
To find the volume of the cone, we need to express its height in terms of the radius of the cylinder's base. The height of the cone will be equal to the height of the cylinder.
Now, let's solve for h in terms of r using the given information:
66 = π * r^2 * h
h = 66 / (π * r^2)
Substituting this value of h into the volume formula of the cone:
Volume of the cone = (1/3) * π * r^2 * (66 / (π * r^2))
Volume of the cone = (1/3) * 66
Volume of the cone = 22
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An element with mass 640 grams decays by 7. 3% per minute. How much of the element is remaining after 8 minutes, to the nearest 10th of a gram?.
The remaining mass after 8 minutes is approximately 382.1 grams.To calculate the remaining mass of the element after a certain number of minutes given the decay rate, we can use the formula:
Remaining mass = Initial mass * (1 - decay rate)^number of minutes
Given:
Initial mass = 640 grams
Decay rate = 7.3% = 7.3/100 = 0.073
Number of minutes = 8
Plugging in these values into the formula, we can calculate the remaining mass:
Remaining mass = 640 * (1 - 0.073)^8
Calculating this expression:
Remaining mass ≈ 640 * (0.927)^8
Remaining mass ≈ 640 * 0.597219683
Remaining mass ≈ 382.086220288
Rounded to the nearest tenth of a gram, the remaining mass after 8 minutes is approximately 382.1 grams.
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For her phone service, Linda pays a monthly fee of $27 and she pays an additional $0.07 per minute of use The least she has been charged in a month is $129.13What are the possible numbers of minutes she has used her phone in a month?
Therefore, the possible numbers of minutes Linda has used her phone in a month are 1459 minutes or more.
To determine the possible number of minutes Linda has used her phone in a month, we can set up an equation based on the given information.
Let's assume the number of minutes Linda has used her phone in a month is represented by 'm'.
The total charge for the phone service consists of the monthly fee of $27 plus the additional charge of $0.07 per minute:
Total charge = $27 + $0.07 * m
According to the given information, the least amount Linda has been charged in a month is $129.13. So we can set up the following equation:
$27 + $0.07 * m ≥ $129.13
Now we can solve this equation to find the possible range of values for 'm'.
$0.07 * m ≥ $129.13 - $27
$0.07 * m ≥ $102.13
m ≥ $102.13 / $0.07
m ≥ 1459
Therefore, the possible numbers of minutes Linda has used her phone in a month are 1459 minutes or more.
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Three vertices of parallelogram WXYZ are W(-5,2), X(2,4), and Z(-7, -3). Find the coordinates of vertex Y
The coordinates of Y are (-1,0). Answer: The coordinates of vertex Y are (-1,0).
To find the coordinates of vertex Y of parallelogram WXYZ whose coordinates are given, we need to use the properties of a parallelogram.
A parallelogram is a quadrilateral in which opposite sides are parallel and congruent.
It means the distance between points W and X is equal to the distance between points Y and Z.
W(-5,2), X(2,4), Y(a,b), and Z(-7,-3).
Therefore, the length of side WX is given as,
WX = √ [(2 - (-5))²+ (4 - 2)²]
= √(49 + 4) = √53. ..(1)
As opposite sides of parallelogram are parallel, XY will have the same slope as WZ. The slope of line WZ is given as
(2 - (-3))/(-5 - (-7)) = 5/2.
Therefore, the slope of XY is also 5/2.
(a, b) is on line XY. The equation of line XY can be written as
y - b = 5/2(x - a) ...(2)
It is given that the length of side WZ is equal to the length of side WX. Therefore,
WZ = WX = √53.
Distance WY can be found using the distance formula as follows:
WY² = WZ² - ZY²WY² = 53 - (b + 3)² ...(3)
Similarly, distance XY can be found as,
XY²= WX²- WY²XY²
= 53 - (a - 2)² - (b - 4)² ...(4)
By solving equations (2), (3), and (4) for a and b, we can get the coordinates of Y.a = -1, b = 0
Therefore, the coordinates of Y are (-1,0). Answer: The coordinates of vertex Y are (-1,0).
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Consider the chance experiment of rolling two fair ten-sided dice and adding the
values shown on the dice. The sides of each die are labeled 0, 1, 2, 3, 4, 5, 6, 7, 8 and
9.
What is the probability that the outcome of this chance experiment results in a sum of
9?
To calculate the probability of rolling two fair ten-sided dice and obtaining a sum of 9, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes. Therefore, the probability of obtaining a sum of 9 when rolling two fair ten-sided dice is 0.1 or 10%.
Let's analyze the possible combinations of rolls that result in a sum of 9. The following combinations can yield a sum of 9: (0, 9), (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1), and (9, 0). There are 10 favorable outcomes.
Since each die has ten sides, there are a total of 10 possible outcomes for each die. Thus, the total number of possible outcomes for rolling two ten-sided dice is 10 * 10 = 100.
To calculate the probability, we divide the number of favorable outcomes (10) by the total number of possible outcomes (100):
Probability = Favorable outcomes / Total outcomes = 10 / 100 = 1/10 = 0.1.
Therefore, the probability of obtaining a sum of 9 when rolling two fair ten-sided dice is 0.1 or 10%.
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what is 360kg as a fraction of 480kg? (as a simplified fraction.)
The simplified fraction of 360kg as a fraction of 480kg is 51/68.
To determine what 360kg is as a fraction of 480kg as a simplified fraction, follow these steps:
Step 1:
Determine the GCD of 360 and 480:
Firstly, let's determine the greatest common divisor of 360 and 480.
The greatest common factor (GCF) is the largest number that divides two or more numbers equally.
We'll use Euclid's algorithm to determine the GCD of the two numbers.
480 = 360 × 1 + 120360
= 120 × 3 + 0260
= 120 × 2 + 247
= 120 × 2 + 7
The GCD is 7, according to Euclid's algorithm.
Step 2:
Simplify the fraction using the GCD: Since the GCD of 360 and 480 is 7, we may simplify the fraction 360/480 by dividing the numerator and denominator by 7 to get the simplified fraction of 360/480 ÷ 7/7, which equals 51/68.
Hence, the simplified fraction of 360kg as a fraction of 480kg is 51/68.
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k/3+3less than or equal to sign - 2
SOLVE FOR K
Given inequality is `k/3 + 3 ≤ -2`.Solve for k.
Step 1: Subtract 3 from both sides of the inequality to isolate k/3.
k/3 ≤ -2 - 3
k/3 ≤ -5
Step 2: Multiply both sides of the inequality by 3 to isolate k.
k ≤ -5 * 3
k ≤ -15
Hence, the solution of the given inequality `k/3 + 3 ≤ -2` is k ≤ -15.
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CERAMICS Josh has 8 days to make pots and plates to sell at a local fair. Each potweighs 2 pounds and each plate weighs 1 pound. Josh cannot carry more than 50 poundsto the fair. Each day, he can make at most 5 plates and at most 3 pots. He will make $12profit for every plate and $25 profit for every pot that he sells.a. Write linear inequalities to represent the number of pots p and plates a Josh maybring to the fair.b. List the coordinates of the vertices of the feasible region.c. How many pots and how many plates should Josh make to maximize his potentialprofit?
The given restrictions can be written as follows:Maximum weight carried by Josh: 2p + 1a ≤ 50Maximum number of plates per day: p ≤ 3his objective function would be:Profit = 12a + 25pWe need to find the values of a and p which can maximize his profit.
Thus, the linear inequalities to represent the number of pots p and plates a that Josh may bring to the fair is:2p + 1a ≤ 50, a ≤ 5 and p ≤ 3.b) The feasible region can be found by plotting the given constraints on the coordinate plane. Here is the graph for the same:From the graph, we can see that the vertices of the feasible region are (0,0), (3,5), (8,0), and (16,0).c) Josh wants to maximize his profit.
Therefore, To do so, we can substitute the vertices of the feasible region and calculate the profit to identify the combination of pots and plates that gives the maximum profit.
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It costs the developer $300,000 to build each townhouse and $450,000 to build each single-family home. Write a function that can be used to determine the minimum cost.
The function for determining the minimum cost of townhouse and single-family home development is min_cost = (num_townhouses x 300000) + (num_homes x 450000).
A function is a self-contained block of code that performs a specific task. In the given problem, we need to determine the minimum cost of developing townhouses and single-family homes. Here, the cost of building a townhouse is $300,000 while the cost of building a single-family home is $450,000. We need to determine the minimum cost by multiplying the number of townhouses and single-family homes by their respective costs.
Therefore, the function for determining the minimum cost of townhouse and single-family home development is given by: min_cost = (num_townhouses x 300000) + (num_homes x 450000) where num_townhouses and num_homes are the number of townhouses and single-family homes, respectively. This function takes two arguments and returns the minimum cost for developing the given number of townhouses and single-family homes.
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Matt made the model below to help him solve math problem Complete the expression that matches matt's model
The expression that matches Matt's model is 1/3 × 3/4.
From the given model,
A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
The expression to represent the model
1/3 × 3/4
= 1/4
Therefore, the expression that matches Matt's model is 1/3 × 3/4.
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16 oz = 1 lb
it says to write it in two unit multipliers.
The two-unit multipliers for the conversion of 16 oz to 1 lb are as follows are 16 oz / 1 lb and 1 lb / 16 oz
Unit multiplier refers to a way to convert one unit of measurement to another unit of measurement by multiplying it by a ratio of two equivalent units. This ratio is known as the conversion factor. A unit multiplier can be used to convert from one unit to another because it represents the relationship between the two units of measure.
There are different methods to convert between units of measure. One of these methods is the use of unit multipliers. The use of unit multipliers is an effective way of converting between units because it is straightforward and consistent with the basic principles of mathematics.
The two-unit multipliers for the conversion of 16 oz to 1 lb are 16 oz / 1 lb and 1 lb / 16 oz. The first ratio represents the number of ounces in one pound, while the second ratio represents the number of pounds in one ounce. To convert 16 oz to pounds, we multiply by the ratio of 1 lb / 16 oz. To convert 1 lb to ounces, we multiply by the ratio of 16 oz / 1 lb.
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Which expressions are equivalent to 8. 9 x 6. 2 8. 7? Check all that apply. 9 x 6 9 8. 9 6. 2 8. 7 x 8. 9 x 8. 7 6. 2 8. 7 8. 9 x 6. 2 6. 2 8. 7 8. 9 6. 2 8. 7 8. 9 x 8. 9 6. 2 x 8. 7.
The following expressions are equivalent to 8.9 x 6.28.7: 9 x 6; 6.28.7; 8.9 x 6.2. The product of two numbers, in general, is the outcome when we multiply the numbers together.
It means, when we take two quantities and multiply them, we get the result as a product. Let us understand how the multiplication of numbers works with an example. When we multiply 3 and 4, we get:3 × 4 = 12Here, 3 and 4 are called factors, and the result, 12, is called the product. Equivalent expressions are the expressions that have the same value, but their structures may differ. The expressions can be equivalent if they have the same value, but their format is different .Let's list the expressions that are equivalent to 8.9 x 6.28.7:The product of 9 and 6 is equal to 54. The product of 8.9 and 6.2 is equal to 55.18.The expression 6.28.7 is the same as 55.18. The product of 8.9 and 6.2 is the same as 6.28.7.Therefore, the following expressions are equivalent to 8.9 x 6.28.7:9 x 66.28.78.9 x 6.2.
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What is another important math contribution fibonacci made.
Another important mathematical contribution made by Fibonacci (Leonardo of Pisa) is the introduction of the Hindu-Arabic numeral system and its adoption in Western mathematics.
During the 13th century, Fibonacci traveled extensively and encountered the numeral system used in India and the Arab world, which is based on the concept of place value. Recognizing its efficiency and advantages over the existing Roman numeral system, Fibonacci popularized the Hindu-Arabic numeral system in Europe through his influential book "Liber Abaci" (Book of Calculation) published in 1202.
The Hindu-Arabic numeral system, which includes the use of the digits 0-9 and the concept of positional notation, revolutionized arithmetic and made calculations much simpler and more efficient. It introduced the concept of using place value to represent numbers, allowing for complex calculations to be performed with ease. This numeral system formed the foundation of modern arithmetic and laid the groundwork for the development of algebra and other branches of mathematics.
Fibonacci's introduction and promotion of the Hindu-Arabic numeral system had a profound impact on mathematics, commerce, and scientific endeavors. It enabled advancements in fields such as astronomy, engineering, and finance, making complex calculations more accessible and accurate. Fibonacci's contribution played a crucial role in the development of mathematics and its practical applications, and it remains a significant part of our everyday numerical representation and calculation systems today.
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Connie has two credit cards, U and V. Card U has a balance of $414. 55, and Card V has a balance of $751. 81. The minimum monthly payment on Card U is 2. 82% of the balance, and the minimum monthly payment on Card V is 3. 09% of the total balance. How much greater is the minimum payment on Card V than on Card U? a. $8. 70 b. $10. 73 c. $9. 51 d. $11. 54 Please select the best answer from the choices provided A B C D.
The minimum payment on Card V is approximately $11.51 greater than the minimum payment on Card U. Among the given choices (a. $8.70, b. $10.73, c. $9.51, d. $11.54), the closest answer to $11.51 is option c. $9.51.
To find the difference in the minimum payments between Card V and Card U, we need to calculate the minimum payment for each card and then subtract the minimum payment of Card U from Card V. First, let's calculate the minimum payment for Card U: Minimum payment on Card U = 2.82% of $414.55 = (2.82/100) * $414.55 ≈ $11.68
Next, let's calculate the minimum payment for Card V: Minimum payment on Card V = 3.09% of $751.81= (3.09/100) * $751.81≈ $23.19. Finally, we can find the difference between the minimum payments: Difference = Minimum payment on Card V - Minimum payment on Card U= $23.19 - $11.68≈ $11.51.
Therefore, the minimum payment on Card V is approximately $11.51 greater than the minimum payment on Card U. Among the given choices (a. $8.70, b. $10.73, c. $9.51, d. $11.54), the closest answer to $11.51 is option c. $9.51.
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Name two characteristics of nonvertical lines that make slope-intercept form (y = mx + b) a good choice when writing an equation for this type of line.
Two characteristics of nonvertical lines that make slope-intercept form (y = mx + b) a good choice when writing an equation for this type of line are:
Slope: The slope of a line represents the rate of change between the y-coordinates and x-coordinates. In slope-intercept form, the slope (m) is explicitly represented as a coefficient of x. This allows us to easily determine the steepness and direction of the line. By knowing the slope, we can understand how the line is changing and make predictions about its behavior.
y-intercept: The y-intercept (b) in slope-intercept form represents the value of y when x is equal to zero. It indicates the point where the line crosses the y-axis. Having the y-intercept explicitly stated in the equation allows us to quickly identify the starting point of the line and understand its initial position on the coordinate plane.
By having both the slope and y-intercept explicitly defined in the equation, slope-intercept form provides valuable information about the line's behavior, direction, and starting position. It simplifies the process of graphing and understanding the line's characteristics.
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The shape shown is made up of three similar right-angled triangles.
Click to insert IMC 2022 KF3a
The smallest triangle has two sides of side-length 2, as shown.
What is the area of the shape?
To calculate the area of the shape made up of three similar right-angled triangles, we need additional information about the scale factor or proportions of the triangles. Without that information, we cannot determine the exact area of the shape.
The given information states that the shape is composed of three similar right-angled triangles, and the smallest triangle has two sides of side-length 2. While we know the dimensions of the smallest triangle, we do not have any information about the scale factor or proportions of the other two triangles. Since the shape is formed by three similar triangles, the areas of the triangles would be proportional, but we cannot determine the exact proportions without additional information. Consequently, we cannot calculate the area of the shape accurately based solely on the given information.
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ken can work at most 12 hours next week He needs too earn at least $80 to cover his gas and food expenses. He earns $10 per hour in a supermarket and $5 per hour in a farm. Let x be the number of hours he works in the supermarket and y be the number of hours he works in the farm, write a system of linear inequlaties to model the situaition then solve
Given that Ken can work at most 12 hours next week, he needs to earn at least $80 to cover his gas and food expenses. He earns $10 per hour in a supermarket and $5 per hour in a farm.
Let x be the number of hours he works in the supermarket and y be the number of hours he works in the farm. We need to write a system of linear inequalities to model the situation.Linear inequality to model the situation will be:x + y ≤ 12 ---(1) [Ken can work at most 12 hours next week]10x + 5y ≥ 80 ---(2) [Ken needs to earn at least $80 to cover his gas and food expenses]Thus, the required system of linear inequalities is[tex]:x + y ≤ 12 (1)10x + 5y ≥ 80[/tex] (2)Now, we need to solve the system of linear inequalities to find the feasible solutions. We will solve the inequalities using the method of graphing.Linear Inequality (1)[tex]:x + y ≤ 12x + y = 12y = -x + 12[/tex]The graph of the inequality y = -x + 12 is shown below:Graph of inequality y = -x + 12:Let's test the point (0, 12) in the inequality x + y ≤ 12:0 + 12 ≤ 12⇒ 12 ≤ 12This is true. So, the solution to this inequality is below or on the line y = -x + 12.
Linear Inequality (2):10x + 5y ≥ 8010x + 5y/5 ≥ 80/5⇒ 2x + y ≥ 16y ≥ -2x + 16The graph of the inequality y ≥ -2x + 16 is shown below:Graph of inequality y ≥ -2x + 16:Let's test the point (0, 16) in the inequality 2x + y ≥ 16:2(0) + 16 ≥ 16⇒ 16 ≥ 16This is true. So, the solution to this inequality is above or on the line y = -2x + 16.Thus, the feasible solutions are the region in the graph where both the inequalities overlap and hence, are satisfied. The shaded region in the graph below represents the feasible region. The points on the line are also included.Feasible region:Let's solve for the points of intersection of the lines y = -x + 12 and
y = -2x + 16:y
= -x + 12y
= -2x + 16
⇒ -x + 12 = -2x + 16
⇒ x = 4y = -x + 12
⇒ y = 8
Thus, the point of intersection of the two lines is (4, 8).So, the solution is (x, y) = (4, 8). Therefore, Ken should work for 4 hours in the supermarket and 8 hours in the farm to earn at least $80 to cover his gas and food expenses.
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A system of linear inequalities to model the situation is given by:
x + y ≤ 12
10x + 5y ≥ 80
A possible solution for this system of linear inequalities is (4, 8).
How to write a system of inequalities to model this situation?In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of hours Ken works in the supermarket and the number of hours Ken works in the farm respectively, and then translate the word problem into a linear inequality as follows:
Let the variable x represent the number of hours Ken works in the supermarket.Let the variable y represent the number of hours Ken works in the farm.Since Ken would work at most 12 hours while earning $10 per hour in a supermarket and $5 per hour in a farm, and he needs too earn at least $80, a system of linear inequalities that models the situation and constraints is given by;
x + y ≤ 12
10x + 5y ≥ 80
By solving the system of linear inequalities, we have:
10(12 - y) + 5y ≥ 80
120 - 10y + 5y ≥ 80
120 - 5y ≥ 80
5y ≥ 120 - 80
5y ≥ 40
y ≥ 40/5
y ≥ 8
For the value of x, we have:
x ≤ 12 - y
x ≤ 12 - 8
x ≤ 4
In conclusion, a possible solution (x, y) is (4, 8).
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The base of the paperweight has an area of 112.5 sq cm and the height of the paperweight is 11 cm.What is the volume of the paperweight?
Given that the base has an area of 112.5 square centimeters and the height is 11 centimeters, the volume is calculated as 1237.5 cubic centimeters.
To find the volume of the paperweight, we can use the formula for the volume of a cylinder, as a paperweight can be approximated as a cylinder. The formula for the volume of a cylinder is given by:
Volume = Base Area × Height
Given that the base of the paperweight has an area of 112.5 square centimeters and the height of the paperweight is 11 centimeters, we can substitute these values into the formula to calculate the volume:
Volume = 112.5 sq cm × 11 cm
To find the volume, we multiply the values together:
Volume = 1237.5 cubic centimeters
Therefore, the volume of the paperweight is 1237.5 cubic centimeters.
In order to determine the volume of the paperweight, we consider its three-dimensional shape. A paperweight can be approximated as a cylinder, which consists of a circular base and a height.
Given that we have the area of the circular base, which is 112.5 square centimeters, and the height of the paperweight, which is 11 centimeters, we can use the formula for the volume of a cylinder to find the total amount of space enclosed by the paperweight.
The formula states that the volume of a cylinder is equal to the product of the base area and the height. By substituting the given values into the formula, we can calculate the volume of the paperweight.
In this case, multiplying the base area of 112.5 square centimeters by the height of 11 centimeters gives us a volume of 1237.5 cubic centimeters.
This means that the paperweight occupies a total space of 1237.5 cubic centimeters within its cylindrical shape.
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A surveyor can measure the width of a river by setting up a transit (surveying device) at a point Con
one side of the river and taking a sighting of a point A on the other side. After turning through an
angle of 90 degrees at C, the surveyor walks a distance of 150 feet to point B. Using the transit at B,
the angle is measured and found to be 35°. What is the width of the river rounded to the nearest
meter?
0-35
150 ft
To measure the width of the river, the surveyor uses trigonometry. With an angle of 35° at point B, a 90° turn at C, and a 150 ft distance, the width of the river is approximately 76 meters.
To find the width of the river, we can use trigonometry. Let's assume the width of the river is represented by the letter 'x'.
Given that the surveyor turned 90 degrees at point C, we can conclude that angle A is also 90 degrees. Using the given angle at point B (35 degrees), we can calculate angle C as 180 - 90 - 35 = 55 degrees.Now, we have a right-angled triangle with angle C as 55 degrees, side BC as 150 feet, and we want to find side AC (which represents the width of the river).
Using the trigonometric ratio tangent, we can write the equation:
tan(55 degrees) = AC / BC
Solving for AC:
AC = tan(55 degrees) * BC
= tan(55 degrees) * 150 ft
Now, we can convert the distance from feet to meters. Assuming 1 foot is approximately 0.3048 meters, we have:
AC = tan(55 degrees) * 150 ft * 0.3048 m/ft
≈ 75.69 m
Rounded to the nearest meter, the width of the river is approximately 76 meters.
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Choose the division equation that represents the fraction shown.one fourteenth = ________
The division equation that represents the fraction one fourteenth is 1 ÷ 14 = 0.0714. This equation shows the process of dividing 1 into 14 equal parts, resulting in a value of approximately 0.0714 for each part.
To find the division equation that represents the fraction one fourteenth, we need to determine the relationship between the numerator and the denominator. In this case, the numerator is 1, and the denominator is 14. Therefore, the division equation can be written as:
1 ÷ 14 = x
where "x" represents the unknown value we are trying to solve.
The fraction one fourteenth represents the idea of dividing 1 into 14 equal parts. In mathematics, division is the process of distributing a quantity (the numerator) into equal parts (the denominator). The numerator is divided by the denominator to determine the value of each part.
In this case, 1 is the quantity being divided, and 14 is the number of equal parts we want to divide it into. So, we can write the division equation as 1 ÷ 14.
To find the value of x, we need to solve the division equation. Dividing 1 by 14 gives us a decimal value:
1 ÷ 14 ≈ 0.0714
Therefore, the division equation that represents the fraction one fourteenth is 1 ÷ 14 = 0.0714. This means that each part, when 1 is divided into 14 equal parts, has a value of approximately 0.0714.
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Milo wants to make a mixture that is 50% lemon juice and 50% lime juice. How much 100% lemon juice should he add to a juice mixture that is 20% lemon juice and 80% lime juice to make 4 gallons of the 50% lemon/50% lime juice mixture? 0. 5 gallon 1. 5 gallons 2 gallons 2. 5 gallons.
To solve this problem, we can set up an equation based on the volume of lemon juice in the mixture:
Let's assume Milo needs to add x gallons of 100% lemon juice.
The total volume of the final mixture is given as 4 gallons, and it should be a 50% lemon juice and 50% lime juice mixture.
The initial mixture contains 20% lemon juice, which means it contains 20% of 4 gallons = 0.2 * 4 = 0.8 gallons of lemon juice.
So, the equation becomes:
0.8 gallons (initial lemon juice) + x gallons (additional 100% lemon juice) = 0.5 * 4 gallons (final lemon juice)
Simplifying the equation:
0.8 + x = 2
Subtracting 0.8 from both sides:
x = 2 - 0.8
x = 1.2
Therefore, Milo needs to add 1.2 gallons of 100% lemon juice to make 4 gallons of the 50% lemon/50% lime juice mixture.
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7-1 reteach to build understanding Adding and Subtracting Polynomials
Adding and subtracting polynomials is a fundamental concept in algebra. It involves combining like terms and following the rules of operations to simplify expressions.
Adding and subtracting polynomials involves combining similar terms and following the rules of operations. To add or subtract polynomials, we align like terms and perform the indicated operations. Like terms are terms that have the same variables raised to the same powers. For example, in the expression 3[tex]x^{2}[/tex] + 2x - 5 + 4[tex]x^{2}[/tex] - 3x + 2, we can group the like terms together: (3[tex]x^{2}[/tex] + 4[tex]x^{2}[/tex]) + (2x - 3x) + (-5 + 2). By combining like terms, we obtain 7x^2 - x - 3.
Then subtracting polynomials, we can think of it as adding the opposite. For instance, to subtract 2[tex]x^{2}[/tex] - 3x + 4 from 5[tex]x^{2}[/tex]+ 2x - 6, we change the signs of the second polynomial and then add the two polynomials together: (5[tex]x^{2}[/tex] + 2x - 6) + (-2[tex]x^{2}[/tex] + 3x - 4). We combine like terms to simplify the expression: (5[tex]x^{2}[/tex]- 2[tex]x^{2}[/tex]) + (2x + 3x) + (-6 - 4). This results in 3[tex]x^{2}[/tex] + 5x - 10.
It is essential to follow the rules of operations, such as adding or subtracting coefficients and keeping the variable part unchanged. By carefully aligning like terms and performing the indicated operations, we can simplify expressions involving the addition and subtraction of polynomials effectively.
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Which of the following shows 9x2y − 4x 3y3x − 2y2 written in standard form? 9x2y − 4x 3y3x − 2y2 3y3x − 2y2 9x2y − 4x 9x2y − 4x − 2y2 3y3x 3y3x 9x2y − 2y2 − 4x.
Therefore, the expression 9x²y - 4x + 3y³x - 2y² written in standard form is 3y³x + 9x²y - 2y² - 4x.
To express the expression 9x²y - 4x + 3y³x - 2y² in standard form, we need to combine like terms.
The expression can be rewritten as:
9x²y + 3y³x - 4x - 2y²
The standard form arranges the terms in descending order of the exponents of the variables. So, let's rearrange the terms:
3y³x + 9x²y - 2y² - 4x
Therefore, the expression 9x²y - 4x + 3y³x - 2y² written in standard form is 3y³x + 9x²y - 2y² - 4x.
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The music for Savannah’s dance routine lasts for exactly 4 minutes. When Savannah dances
her routine, she starts with her music and finishes 12 seconds before the music ends.
What percent of the time the music is playing is Savannah dancing?
The answer is that Savannah is dancing 95% of the time the music is playing. Duration of music = 4 minutes Duration of Savannah's dance routine = 4 - (12/60) = 3.8 minutes. Now, we need to find the percentage of time the music is playing is Savannah dancing.
To find the percentage of time, we need to divide the time for Savannah's dance routine by the duration of the music and then multiply the quotient by 100.Percentage of time Savannah is dancing = (time for Savannah's dance routine / duration of music) × 100= (3.8 / 4) × 100= 95%.
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