subtract and reduce to lowest terms of 4 5/7- 2 4/7

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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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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The surface area of the prism is 252in²

Given,

Dimensions of rectangular prism:

Height = 2 in

width = 8 in

depth = 11 in

Now,

The surface area of a rectangular prism is given by:

A = 2 * (x * y) + 2 * (x * z) + 2 * (y * z)

where,

x, y, z: are the sides of the rectangular prism

Substituting values we have:

A = 2 * (2 * 8) + 2 * (2 * 11) + 2 * (8 * 11)

A = 252 in²

Hence the surface area of the prism in square inches will be 252 .

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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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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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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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The period of the FM wave is 1.13 × 10–8 seconds.

The correct option is B. 1. 13 × 10–8 seconds.

How to find the period of an FM wave.

The period of a wave can be defined as the time required to complete one cycle.

The period is represented by the symbol T.

It is measured in seconds (s).

Frequency is the number of cycles per second. It is measured in Hertz (Hz).

It is represented by the symbol f.

The formula for finding the period of a wave is given by:

T = 1/f

Given that the frequency of a FM wave is 8.85 x 10⁷ Hertz.

We have to find the period of the FM wave.

The period of the FM wave can be found using the formula:

T = 1/f

Where frequency

= 8.85 × 107 Hz

T = 1/fT

= 1/8.85 x 10⁷T

= 1.13 × 10–8 seconds

Therefore, the period of the FM wave is 1.13 × 10–8 seconds.

Thus, option B is the correct answer.

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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 coordinates of the fourth vertex are (1.1, -3.1).

We know that a rectangular piece of material has two pairs of parallel sides and four right angles. Therefore, we can calculate the distance between two opposite sides by using the distance formula. The two sides that are parallel to the x-axis have the same y-coordinate, and the two sides parallel to the y-axis have the same x-coordinate.

Thus, the distance between opposite sides is given by the difference between the x-coordinates or the difference between the y-coordinates. The x-coordinates of the vertices are −1.3, −1.3, and 1.1. So the difference between them is 1.1 − (−1.3) = 2.4. The y-coordinates of the vertices are −3.1, 1.2, and 1.2. So the difference between them is 1.2 − (−3.1) = 4.3. The fourth vertex is located at the point with an x-coordinate of 1.1 and a y-coordinate of −3.1. Therefore, the coordinates of the fourth vertex are (1.1, -3.1).

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The fact family 12, 4, 3, 2, and 9 demonstrates the relationship between addition and subtraction. In this fact family, when we subtract 3 from 4, the result is 1. If we then add 2 to 1, we get 3. Furthermore, when we subtract 2 from 9, the result is 7. If we add 7 and 2 together, the sum is 9.



In the given fact family, the numbers 4 and 3 are related through subtraction. When we subtract 3 from 4, we get 1. This is represented by the equation 4 - 3 = 1.

On the other hand, the numbers 2 and 1 are related through addition. If we add 2 to 1, the sum is 3. This is represented by the equation 1 + 2 = 3.

Similarly, the numbers 9 and 2 are related through subtraction and addition. When we subtract 2 from 9, we get 7, represented by the equation 9 - 2 = 7. By adding 7 and 2 together, the sum is 9, represented by the equation 7 + 2 = 9.

Therefore, the fact family 12, 4, 3, 2, and 9 demonstrates the relationship between addition and subtraction within the given numbers.

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To find the gradient (or slope) of the line passing through the points (-4, 4) and (-3, 1), we can use the formula: gradient = (change in y) / (change in x).

Let's calculate the change in y and change in x: Change in y = 1 - 4 = -3. Change in x = -3 - (-4) = -3 + 4 = 1. Now we can substitute these values into the formula: gradient = (-3) / (1) = -3.

Therefore, the gradient of the line passing through the points (-4, 4) and (-3, 1) is -3. The negative sign indicates that the line has a negative slope, meaning it is decreasing as we move from left to right on the coordinate plane.

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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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Blorgs have a lifespan of two hours, and after that, they die. The newborn Blorgs continue the reproductive cycle, but all Blorgs born before 3 PM would have reached their lifespan and died by 4:05 PM.

To determine the number of live Blorgs the Mad Professor has at 4:05 PM, we need to understand the pattern of Blorg reproduction and lifespan.

From the given information, we know that one hour after a Blorg is born, it gives birth to a new Blorg. Two hours after birth, it gives birth to two more new Blorgs and then immediately dies.

Let's break down the timeline:

At noon, the Professor has one newborn Blorg.

One hour later, at 1 PM, the initial Blorg gives birth to a new Blorg. So, there are two Blorgs now.

Two hours after birth, at 2 PM, the initial Blorg dies, but the newborn Blorg from 1 PM gives birth to two more Blorgs. So, there are four Blorgs now.

At 3 PM, the two newborn Blorgs from 1 PM give birth to two more Blorgs each, resulting in a total of eight Blorgs.

At 4 PM, the four Blorgs that were born at 2 PM die, while the four Blorgs born at 3 PM are still alive. The four living Blorgs are from the second generation.

Now, let's calculate the number of live Blorgs at 4:05 PM:

At 4 PM, there are four living Blorgs.

Since each Blorg lives for two hours, at 4:05 PM, it has been 5 minutes past their lifespan, and all the living Blorgs from the 3 PM generation would have died.

Therefore, at 4:05 PM, the Mad Professor would have zero live Blorgs.

It is important to note that Blorgs have a lifespan of two hours, and after that, they die. The newborn Blorgs continue the reproductive cycle, but all Blorgs born before 3 PM would have reached their lifespan and died by 4:05 PM.

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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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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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The number of strikeouts in 36 innings by Warren is 45. The number of strikeouts in 80 innings is 96 while the number of strikeouts in 36 innings is 45. To compare the two, we need to know the number of strikeouts by Warren in 36 innings if he pitched for 80 innings.

Warren pitched 80 innings and got 96 strikeouts. The strikeouts he got in 36 innings is 45.So, the difference between the number of innings he pitched in both cases is: 80 - 36 = 44 innings.The number of strikeouts he got for the extra 44 innings is: 96 - 45 = 51 strikeouts.Therefore, Warren has more strikeouts in 80 innings than he has in 36 innings. The number of strikeouts is 96 and the number of innings is 80.

This can be confirmed by subtracting 45 strikeouts in 36 innings from 96 strikeouts in 80 innings:96 - 45 = 51The answer, in more than 100 words, is that Warren had more strikeouts in 80 innings than he had in 36 innings.

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The number of siblings which is least likely can be identified by observing the probabilities given for each possible number of siblings of a randomly selected student.

The least likely number of siblings can be considered as the one that has the smallest probability of being selected.

We can see that the probabilities of having 0 siblings, 1 sibling, 2 siblings, 3 siblings, 4 siblings, and 5 siblings are given below:

The probability of 0 siblings is 0.21.

The probability of 1 sibling is 0.33.

The probability of 2 siblings is 0.24.

The probability of 3 siblings is 0.14.

The probability of 4 siblings is 0.

The probability of 5 siblings is 0.08.

From these probabilities, we can see that the probability of having 4 siblings is 0, which means there is no possibility of randomly selecting a student with exactly 4 siblings. Therefore, having 4 siblings is the least likely scenario based on the given probability distribution.

Therefore, by looking at the graph above, we can see that the least likely number of siblings is 4 siblings.

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The scatter plot shows a comparison between the amount of television watched by students (in hours per week) and their GPA. a reasonable correlation coefficient for these data would be close to 0, indicating a weak or negligible correlation.

Based on the scatter plot, we can observe the general trend of the data points. If the points on the plot are more closely clustered around a straight line, it indicates a stronger correlation between the two variables. Conversely, if the points are more spread out and do not follow a clear pattern, it suggests a weaker correlation. To determine the correlation coefficient, we need to assess the direction and strength of the relationship.

If the data points on the scatter plot exhibit a clear upward or downward trend, it indicates a positive or negative correlation, respectively. The correlation coefficient ranges between -1 and 1, with values closer to -1 or 1 indicating a stronger correlation. A correlation coefficient of 0 indicates no linear relationship. Given that the scatter plot does not show a clear pattern or trend, it suggests a weak or no correlation between the amount of television watched and GPA. Therefore, a reasonable correlation coefficient for these data would be close to 0, indicating a weak or negligible correlation.

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To determine the lengths of QT and RS using the Pythagorean Theorem, we need additional information such as the lengths of the other sides or the measurements of angles in the triangle. Without this information, we cannot accurately calculate the lengths of QT and RS.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. However, in the given question, we are not provided with any measurements or information about angles in the triangle or the lengths of the other sides.

Therefore, without additional information, we cannot determine the lengths of QT and RS using the Pythagorean Theorem.

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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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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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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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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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n = 45 satisfies the equation, and it is the value that makes 3/5 of n equal to 27. The value of n can be determined by setting up an equation based on the given information that 3/5 of n is equal to 27.

By solving the equation, we can find the value of n. Let's set up the equation based on the given information:

(3/5) * n = 27

To solve for n, we need to isolate n on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of 3/5, which is 5/3:

n = 27 * (5/3)

Simplifying the right side of the equation:

n = (27 * 5) / 3

n = 135 / 3

n = 45

Therefore, the value of n is 45. We can confirm this by substituting n = 45 back into the equation:

(3/5) * 45 = 27

(3/5) * 45 = 135/5 = 27

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To find the area of a verandah 1 m wide constructed outside a room 5.5 m long and 4 m wide.

The area of the verandah is 45.5 m².

We need to find the area of the overall rectangular structure (room + verandah) and subtract the area of the room.

Area of the overall rectangular structure = (length + 2 × width) × (width + 1)

Area of the room = length × width

Area of the verandah = Area of the overall rectangular structure - Area of the room.

Given, Length of the room = 5.5 m

the Width of the room = 4 m

Width of the verandah = 1 m.

Area of the overall rectangular structure = (length + 2 × width) × (width + 1)

= (5.5 + 2 × 4) × (4 + 1)

= 13.5 × 5

= 67.5 m²

Area of the room = length × width

= 5.5 × 4

= 22 m²

Area of the verandah = Area of the overall rectangular structure - Area of the room

= 67.5 - 22

= 45.5 m²

Therefore, the area of the verandah is 45.5 m².

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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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Walter had new carpeting installed in his 13-foot by 18-foot family room. He paid $7 per square foot for the carpet and $2 per square foot for the pad under the carpet.

The total area of the room will be A

= l × w= 13 × 18= 234 sq. feet The cost of carpet per square feet is $7. The cost of carpeting the family room is:Total cost of carpeting

= Cost of carpet per sq. ft. × Area of the room

= $7 × 234

= $1638The cost of the pad per square feet is $2.Therefore, the cost of padding the family room is: Total cost of padding

= Cost of pad per sq. ft. × Area of the room

= $2 × 234

= $468The total cost of the carpet and pad will be the sum of the cost of the carpeting and the cost of the padding. Total cost of carpet and pad

= Total cost of carpeting + Total cost of padding

= $1638 + $468

= $2106Hence, the total cost of the carpet and pad is $2106.

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The amount that Brenda would pay for the 8 t- shirts at the same rate would be $ 37. 60.

To determine how much Brenda pays for the package of 8 T-shirts, we need to calculate the price per T-shirt based on Jacob's purchase and then multiply it by 8.

Jacob bought 7 T-shirts for $32.90, so the price per T-shirt can be calculated as:

Price per T-shirt = Total cost / Number of T-shirts

Price per T-shirt = $32.90 / 7

Price per T-shirt = $4.70

The cost of 8 shirts is therefore:

Total cost = Price per T-shirt * Number of T-shirts

Total cost = $4.70 x 8

Total cost = $37.60

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Missing detail is:

Jacob bought 7 T shirts for $32.90 Kyle bought 12 T shirts for $60.00

a) To find the value of b that makes the integral equal to 0, we need to solve the integral:

∫(1 to b) (6x - 9) dx = 0

Applying the fundamental theorem of calculus, we can find the antiderivative of the integrand and evaluate it at the limits of integration:

[3x² - 9x] (from 1 to b) = 0

Substituting the limits, we have:

(3b² - 9b) - (3(1)² - 9(1)) = 0

Simplifying further:

3b² - 9b - 6 = 0

Now we can solve this quadratic equation for b using the quadratic formula or factoring:

b² - 3b - 2 = 0

Factoring the quadratic equation, we get:

(b - 2)(b + 1) = 0

Setting each factor to zero, we find two possible values for b:

b = 2 or b = -1.

However, since b cannot equal 1 (as stated in the problem), the only valid solution is b = 2.

b) To verify our answer, we can sketch a graph of y = 6x - 9 on the interval from x = 1 to x = b = 2. The graph will be a straight line with a slope of 6 and y-intercept at -9.

On the interval [1, 2], the line starts at the point (1, -3) and ends at the point (2, 3). By calculating the area under this line, we find that the integral from 1 to 2 of (6x - 9) equals 0, confirming our previous result.

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Let's analyze Sonia's situation step by step.

A. To determine if Sonia will have enough money, we need to calculate her total earnings and compare it to the expenses.

Sonia works 5 hours a week and earns $11 per hour, so her total earnings in a week can be calculated as:

Total Earnings = Hours Worked per Week × Hourly Rate

             = 5 hours × $11/hour

             = $55

Now let's calculate the total expenses:

Total Expenses = Cost of Dinner + Cost of Necklace

             = $40 + $50

             = $90

Since Sonia's total earnings are $55 and her total expenses amount to $90, she will not have enough money to cover both expenses.

B. The tradeoffs Sonia will have to consider involve prioritizing her spending. She will need to decide between going out with her friends or buying her mom a birthday present. She could choose to spend less on the dinner and find a more affordable gift for her mom, or she could skip the dinner and use the money solely for her mom's gift. Sonia will need to weigh the importance of both activities and make a decision based on her personal values and priorities.

C. The decision on how Sonia should spend her money ultimately depends on her individual circumstances and priorities. She could consider alternative options to reduce expenses, such as having a more budget-friendly dinner or finding a more affordable gift for her mom. Additionally, Sonia could explore creative ways to spend time with her friends without incurring significant expenses, like organizing a potluck or a movie night at home. Ultimately, Sonia should prioritize what matters most to her, whether it's celebrating her mom's birthday or enjoying a fun evening with her friends, while considering her financial limitations.

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