Answer:
Total cost for groceries = ($3.75, $1.09,
$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. add them all). = $ 31.3
the amount she paid= $ 35
balance =$ 3.7
therefore she have enough money
- The table shows the number of
miles Michael ran each day over
the past four days. How many
more miles did he run on day 3
than on day 2? Determine if there
is extra or missing information.
By conducting subtraction we know that Michael ran 5 miles more on day 3 than on day 2.
What is subtraction?The four arithmetic operations are addition, multiplication, division, and subtraction.
Removal of items from a collection is represented by the operation of subtraction.
For instance, in the following image, there are 5 2 peaches, which means that 5 peaches have had 2 removed, leaving a total of 3 peaches.
The number that the other number is deducted from is known as a minuend.
Subtrahend: The amount that needs to be deducted from the minuend is known as a subtrahend.
Difference: A difference is an outcome obtained by deducting a subtractor from a minimum.
So, more miles Michael ran on day 3 than on day 2:
= 7 - 2
= 5 miles
Therefore, by conducting subtraction we know that Michael ran 5 miles more on day 3 than on day 2.
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In a GP the 8th term is 8748 and the 4th
term is 108. Find the sum of the 1st 10 terms.
The first term of the GP is 4 and the common ratio is 3. We can now substitute these values into the formula for the sum of the first 10 terms to get 118096.
What is Geometric Progression?
A progression of numbers with a constant ratio between each number and the one before
Let the first term of the geometric progression be denoted by "a" and the common ratio be denoted by "r".
We know that the 4th term is 108, so we can use the formula for the nth term of a GP to write:
a*r³ = 108 .....(1)
We also know that the 8th term is 8748, so we can write:
a*r⁷ = 8748 .....(2)
To find the sum of the first 10 terms, we can use the formula for the sum of a finite geometric series:
S = a(1 - rⁿ)/(1 - r)
where S is the sum of the first n terms of the GP. We want to find the sum of the first 10 terms, so we plug in n = 10:
S = a(1 - r¹⁰)/(1 - r)
We now have two equations (1) and (2) with two unknowns (a and r). We can solve for a and r by dividing equation (2) by equation (1) to eliminate a:
(ar⁷)/(ar³) = 8748/108
r⁴ = 81
r = 3
Substituting r = 3 into equation (1) to solve for a, we have:
a*3³ = 108
a = 4
Therefore, the first term of the GP is 4 and the common ratio is 3. We can now substitute these values into the formula for the sum of the first 10 terms to get:
S = 4(1 - 3¹⁰)/(1 - 3)
S = 4(1 - 59049)/(-2)
S = 4(59048)/2
S = 118096
Therefore, the sum of the first 10 terms of the GP is 118096.
118096.
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instead of using the values {1, 2, 3, 4, 5, 6} on dice, suppose a pair of dice have the following: {1, 2, 2, 3, 3, 4} on one die and {1, 3, 4, 5, 6, 8} on the other. find the probability of rolling a sum of 6 with these dice. be sure to reduce
Answer:
your answer to the lowest terms
The probability of rolling a sum of 6 with these dice is 1/12.
Mr. Brown's Thrift Shop
Quarter of 2012 Profit (in dollars)
1 $9,841.28
2 $8,957.67
3 $7,429.84
4 $11,095.67
How much total profit did Mr. Brown's store earn in the third and fourth quarters?
A.
$17,298.45
B.
$17,548.65
C.
$18,124.78
D.
$18,525.51
The correct option is D. $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.
Explain about addition?In math, addition is the process of adding two or more integers together. The numbers having added are known as addends, while the outcome of the addition process, or the final response, is known as the sum. It is among the most fundamental mathematical operations we employ on a daily basis.
Quarterly profit for Mr. Brown's Goodwill Store in 2012 (in dollars)
1 $9,841.28
2 $8,957.67
3 $7,429.84
4 $11,095.67
Total profit = profit of 3rd quarter + profit of 4th quarter
Total profit = $7,429.84 + $11,095.67
Total profit = $18,525.51.
Thus, $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.
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Find the volume of the solid generated by revolving the region enclosed by the following curves: a. y=4-2x², y = 0, x = 0, y = 2 through 360° about the y-axis. b. x = √y+9, x = 0, y =1 through 360° about the y-axis.
The volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
How to find the volume?To find the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis, we use the formula:
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to find the equation of the curve that is generated when we rotate y = 4 - 2x² around the y-axis. To do this, we use the formula for the equation of a curve generated by revolving y = f(x) around the y-axis, which is:
x² + y² = r²
where r is the distance from the y-axis to the curve at any point (x, y).
Substituting y = 4 - 2x² into this formula, we get:
x² + (4 - 2x²) = r²
Simplifying, we get:
r² = 4 - x²
Therefore, the radius of the circular cross-sections perpendicular to the y-axis is given by:
r = √(4 - x²)
Now, we can integrate πr²dy from y = 0 to y = 2:
V = ∫[0,2] π(√(4 - x²))²dy
V = ∫[0,2] π(4 - x²)dy
V = π∫[0,2] (4y - y²)dy
V = π(2y² - (1/3)y³)∣[0,2]
V = π(8 - (8/3))
V = (16/3)π
Therefore, the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
b. To find the volume of the solid generated by revolving the region enclosed by the curves x = √y+9, x = 0, and y = 1 through 360° about the y-axis, we use the same formula as in part (a):
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to solve the equation x = √y+9 for y in terms of x:
x = √y+9
x² = y + 9
y = x² - 9
Next, we need to find the equation of the curve that is generated when we rotate y = x² - 9 around the y-axis. Using the same formula as in part (a), we get:
r = x
Now, we can integrate πr²dy from y = 1 to y = 10 (since x = 0 when y = 1 and x = 3 when y = 10):
V = ∫[1,10] π(x²)²dy
V = π∫[1,10] x⁴dy
V = π(1/5)x⁵∣[0,3]
V = π(243/5)
Therefore, the volume of the solid generated by revolving the region.
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Find the Z score that has 48.4% of the distributions area to its left.
Answer:
To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.
Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.
Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:
To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.
Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.
Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:
To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.
Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.
Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:
invNorm(0.484)= 1.96
Therefore, the Z-score that has 48.4% of the distribution's area to its left is approximately 1.96.
.2 In the diagram below, given that XY = 3cm, XZY = 30° and YZ = x, is it possible to solve for x using the theorem of Pythagoras? Motivate your answer. Show Calculations
Sin 30 =3/x
1/2=3/x
x=6
the zeros of f(x)=20x^2 - 19x + 3
The quadratic function's zeros are therefore [tex]x = 1[/tex] and [tex]x = 0.2[/tex] . A degree two polynomial in one or so more variables that is a quadratic function.
What ways in which quadratic function be recognized?Three points are used to determine a quadratic function, which has the form [tex]f(x) = ax2 Plus bx + c.[/tex]
[tex]Sqrt(b2 - 4ac) = [-b sqrt(b)][/tex] Where the quadratic function's coefficients are a, b, and c.
Here, [tex]a = 20[/tex] , [tex]b = -19[/tex] , & [tex]c = 3[/tex] . We obtain the quadratic formula by substituting these values: [tex]x = [-(-19) sqrt((-19)2 - 4(20)(3)] / 2(20) (20)[/tex]
When we condense this phrase, we get:
[tex]x = [19 +/- sqrt(361 - 240)] / 40 x = [19 +/- sqrt(121)] / 40\sx = [19 ± 11] / 40[/tex]
Therefore, The zeros of a quadratic equation [tex]f(x) = 20x2 - 19x + 3[/tex] are as follows: [tex]x = (19 Plus 11) / 40 = 1 and x = (19 − 11) / 40 = 0.2.[/tex]
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Help. I dont understand this math question and need help please and thank you.
Answer:
●B. The numbers -1,0,1 are zeros of multiplicity 1.
Step-by-step explanation:
So first, understand that when you are asked for roots, zeros, solutions, or x-intercepts...all of these, they are essentially asking for the same thing. Roots ARE solutions ARE zeros ARE x-intercepts. Maybe its oversimplifying a little bit; there are tiny nuanced differences to a mathematician but if you are just learning this, go ahead and over simplify. They are all the same. So you set it equal to 0 and solve.
Yes, literally, change y to a 0 and solve. See image.
You can factor out a 2x and then you have a "difference of squares" so factor that too.
see image.
"multiplicity" is a cool word. It just means how many times a number is the answer. It sort of doesn't even apply here. 0, -1, and 1 are the answer just one time each...so multiplicity 1. Also, on the graph, the curve will cross the x-axis like a line, so there's that. (See multiplicity 2 is cooler, because the curve will "bounce" at the x-intercept instead, but that's not happening here)
Anyway, set the problem equal to 0 and solve. Ta-da! You're done! Hope this helps! See image.
The mean time to admit an emergency patient to the Mount Nittany Medical Center is 5 minutes with a standard deviation of 3 minutes. Only trauma patients are admitted to this center. Also, assume that the admission process is in fact the radiography process via an X-Ray machine.
(a) What is the natural coefficient of variation for one patient?
C0 : ______________
(b) If the admission times of patients are independent, what will be the mean and variance of admitting a group of 50 emergency patients? What will be the coefficient of variation of a group of 50 emergency patients?
t0 : ______________ σ02 : ______________ C0 :______________
(c) The X-Ray machine in the center may fail at any time randomly. The time to failure is exponentially distributed with a mean of 80 hours and the repair time is also exponentially distributed with a mean of 4 hours. What will be the effective mean and coefficient of variation of the admission time for a group of 50 trauma patients?
te : ______________ σe 2 :____________ Ce :______________
(d) Determine the variability class of the squared-coefficients of variation in Parts a-c (e.g., low variability, moderate variability, or high variability.)
C0 2(Part a): ______________ C0 2(Part b): ______________ Ce 2(Part c): ______________
(e) In two sentences, describe how the manager of center can improve the inflated effective admission time in Part c?
(a) The natural coefficient of variation for one patient is the ratio of the standard deviation to the mean, expressed as a percentage:
C0 = (standard deviation / mean) x 100% = (3 / 5) x 100% = 60%.
(b) If the admission times of patients are independent, the mean and variance of admitting a group of 50 emergency patients can be calculated as follows:
mean = n x mean time = 50 x 5 = 250 minutes
variance = n x variance of individual patient / sample size = 50 x (3)^2 / 50 = 9
The coefficient of variation for a group of 50 emergency patients is the ratio of the standard deviation to the mean, expressed as a percentage:
C0 = (standard deviation / mean) x 100% = (3 / 5) x 100% = 60%.
(c) The effective mean and coefficient of variation of the admission time for a group of 50 trauma patients can be calculated using the following formula:
te = n x mean time / (1 - p1 x p2)
where p1 is the probability of machine failure and p2 is the probability of repair completion. Assuming the machine can fail at any time, p1 can be calculated as 1 / (mean time between failures / mean admission time) = 1 / (80 x 60 / 5) = 0.001042. Assuming the repair time is also exponentially distributed, p2 can be calculated as 1 / mean repair time = 1 / 4 = 0.25. Therefore, te = 50 x 5 / (1 - 0.001042 x 0.25) = 250.14 minutes. The variance of the admission time can be calculated using the formula:
σe^2 = n x variance of individual patient / (1 - p1 x p2)^2 = 50 x (3)^2 / (1 - 0.001042 x 0.25)^2 = 10.81. The coefficient of variation for a group of 50 trauma patients is the ratio of the standard deviation to the mean, expressed as a percentage:
Ce = (standard deviation / mean) x 100% = (sqrt(10.81) / 250.14) x 100% = 2.60%.
(d) The variability class of the squared coefficients of variation can be determined as follows:
C0² (Part a): (0.6)^2 = 0.36 (low variability)
C0² (Part b): (0.6)^2 = 0.36 (low variability)
Ce² (Part c): (0.026)^2 = 0.000676 (low variability)
(e) The manager of the center can improve the inflated effective admission time in Part c by implementing preventive maintenance measures to reduce the probability of machine failure, such as regular inspection and cleaning of the X-Ray machine, and by improving the repair process to reduce the mean repair time, such as hiring more skilled technicians or improving the repair procedures.
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Question 11 (1 point)
(06.03 LC)
What is the product of the expression, 5x(x2)?
a
25x2
b
10x
c
5x3
d
5x2
The expressiοn 5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³). Thus, οptiοn (c) 5x3 is the cοrrect respοnse.
Hοw are prοducts οf expressiοn determined?The cοefficients (the numbers in frοnt οf the variables) οf the expressiοn 5x(x²) can be multiplied, and the expοnents οf the variables can be added, tο determine the prοduct.
The first cοefficient we have is 5 times 1, giving us 5. Sο, using the secοnd x², we have x tο the pοwer οf 2 multiplied by x tο the pοwer οf 1 (frοm the first x). Expοnents are added when variables with the same base are multiplied. Sο, x¹ multiplied by x² results in x³.
Cοmbining all οf the parts, the phrase becοmes:
5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³).
Thus, οptiοn (c) 5x³ is the cοrrect respοnse.
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What was your recommended intake of carbohydrates (grams), and how far were you from it? Show the mathActual Intake Recommended Intake Percentage159.00 115-166 100%
The actual intake of carbohydrates is 138% as compare to recommended intake.
Recommended intake of carbohydrates or any other nutrient are,
Based on the information provided,
Consumed 159 grams of carbohydrates,
Recommended intake is between 115 and 166 grams.
Calculate the percentage of actual intake compared to the recommended intake, use the following formula,
Percentage = (Actual Intake / Recommended Intake) x 100%
Substituting the values in the formula we have,
⇒Percentage = (159 / 115) x 100%
⇒Percentage ≈ 138.3%
Therefore, the actual intake of carbohydrates is about 138% of the recommended intake, indicating that consumption of more carbohydrates than recommended.
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Can someone actually see if I got this correct for this answer please
Your total grade point average (GPA) for the semester is 2.67.
What is GPA ?GPA stands for Grade Point Average. It is a numerical calculation used to measure the academic success of a student. It is calculated by taking the average of all grades received by a student across all courses taken in a given semester or academic year. Each course is assigned a specific number of credit hours, and each grade is assigned a numerical value based on the school’s grading scale. The numerical value of each grade is then multiplied by the number of credit hours for the course, and the total of all courses is added together to determine a student’s GPA.
A higher GPA is generally indicative of higher academic performance while a lower GPA is generally indicative of lower academic performance. A student’s GPA is used by schools, employers, and other organizations to evaluate a student’s academic record.
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This would give you a total of 29.0. Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.
What is GPA?GPA stands for Grade Point Average. It is an academic measure of a student's performance in a course or program of study. It is calculated by dividing the total number of grade points earned by the total number of credit hours taken. A grade point average is typically expressed as a number on a 4.0 scale. A 4.0 GPA is considered to be the highest possible grade point average, while anything below a 2.0 GPA is usually considered to be failing.
This is calculated by adding up the total number of credit hours (10 credit hours) and then multiplying each grade by the respective number of credit hours. A=4.0, B=3.0, C=2.0.
So, you would multiply 4.0 by 3 for FYE 105, 3.0 by 3 for ENG 101, 2.0 by 3 for MAT 150, 2.0 by 3 for BIO 112, and 4.0 by 1 for BIO 113.
This would give you a total of 29.0.
Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.
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A, B, and C are mutually exclusive.
P(A) = .2, P(B) = .2, P(C) = .3. Find P(A ∪ B ∪ C).
P(A ∪ B ∪ C) =
The probability of the union of events A, B, and C is 0.7 where P(A)=0.2, P(B)=0.2 and P(C)=0.3.
What is union?In set theory, the union of two or more sets is a set that contains all the distinct elements of the sets being considered. Formally, the union of sets A and B, denoted as A ∪ B, is the set of all elements that are in either set A or set B, or in both.
According to question:When A, B, and C are mutually exclusive events, it means that they cannot happen at the same time. Therefore, the probability of the union of these events is equal to the sum of their individual probabilities.
In this case, we are given that:
P(A) = 0.2
P(B) = 0.2
P(C) = 0.3
To find the probability of the union of these events, we need to add their probabilities:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
Substituting the given probabilities, we get:
P(A ∪ B ∪ C) = 0.2 + 0.2 + 0.3
P(A ∪ B ∪ C) = 0.7
Therefore, the probability of the union of events A, B, and C is 0.7.
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The question is A, B, and C are mutually exclusive. P(A) = 0.2, P(B) = 0.2, P(C) = 0.3. Find P(A ∪ B ∪ C).
P(A ∪ B ∪ C) = ?
What is the meaning of "Euclidean geometry"?
The concept of Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different theorems and axioms.
What is the concept of Euclidean geometry?The concept of Euclidean geometry as required to be discussed is basically introduced for flat surfaces or plane surfaces. The postulates of the Euclidean geometry are as follows!
1 : A straight line may be drawn from any one point to any other point.
2 :A terminated line can be produced indefinitely.
3 : A circle can be drawn with any centre and any radius.
4 : All right angles are equal to one another (Congruent).
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3.1 Mrs Gilfillan owns a coffee shop. She serves a mixed berry and almond polenta cake that is baked in espresso cups at her coffee shop. She uses the recipe below to make the cake. Mixed Berry and Almond Polenta Calce Makes 15 espresso cups Ingredients 140 g butter 140 g castor sugar 140 g ground almonds 250 g fat-free cottage cheese 75 g mixed frozen berries 25 g polenta 6 eggs separated (keep the yolks for mayonnaise or scrambled egg) Bake at 356 °F until light brown, 30 to 40 minutes. Fat-free cottage cheese is sold in quantities of 125g at R8,99. Calculate the cost of the fat-free cottage cheese required in the recipe.
The cost of the fat-free cottage cheese required in the recipe is R17.98.
How to determine the cost ?
To determine the cost, we first need to calculate the amount of fat-free cottage cheese required in the recipe. The recipe calls for 250g of fat-free cottage cheese, which can be obtained by using 2 units of 125g each. Knowing the cost of ingredients is important for Mrs Gilfillan to price the cake appropriately to cover her costs and make a profit.
What is the cost of the fat-free cottage cheese required to make the Mixed Berry and Almond Polenta Cake recipe that serves 15 espresso cups at Mrs Gilfillan's coffee shop?
The cost of 1 unit of 125g fat-free cottage cheese is R8.99.
Therefore, the cost of 2 units of 125g is calculated as:
R (2 x 8.99) =R 17.98
Hence, the cost of the fat-free cottage cheese required in the recipe is R 17.98.
This cost is in addition to the cost of the other ingredients, such as butter, sugar, ground almonds, mixed frozen berries, and polenta, as well as the cost of labor, overhead, and other expenses involved in making and selling the cake.
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are the ratios 2:1 and 20:10 equivalent
Yes, there is an analogous ratio between 2:1 and 20:10.
What ratio is similar to 2 to 1?We just cancel by a common factor. So 4:2=2:1 . The simplest representation of the ratio 4 to 2 is the ratio 2 to 1. Also, since each pair of numbers has the same relationship to one another, the ratios are equivalent.
By dividing the terms of each ratio by their greatest common factor, we may simplify both ratios to explain why.
As the greatest common factor for the ratio 2:1 is 1, additional simplification is not necessary.
The greatest common factor for the ratio 20:10 is 10. When we multiply both terms by 10, we get:
20 ÷ 10 : 10 ÷ 10
= 2 : 1
As a result, both ratios have the same reduced form, 2:1, making them equal.
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I’ll give brainliest don’t solve just check!
Answer:
Yep
Step-by-step explanation:
Yep. Checking this, all of these are correct. Range on parabolas and absolutes will always go to infinity. Nice work
What percent of 28 is 77?
Answer:
36.3636364%
or 36.36
Step-by-step explanation:
If a₁ = 5 and an
5an-1 then find the value of a4.
If a₁ = 5 and an 5an-1 then The value οf a₄ is 625.
What is arithmetic sequence?An arithmetic sequence is a sequence οf numbers in which each term after the first is fοund by adding a fixed cοnstant number, called the cοmmοn difference, tο the preceding term. Fοr example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a cοmmοn difference οf 3, since each term after the first is fοund by adding 3 tο the preceding term.
The nth term οf an arithmetic sequence can be fοund using the fοrmula:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, and d is the cοmmοn difference. The sum οf the first n terms οf an arithmetic sequence can be fοund using the fοrmula:
Sn = n/2 (a1 + an)
We are given that a₁ = 5, and that the nth term is 5 times the (n-1)th term. We can use this infοrmatiοn tο find the value οf a₄ as fοllοws:
a₂ = 5a₁ = 5(5) = 25
a₃ = 5a₂ = 5(25) = 125
a₄ = 5a₃ = 5(125) = 625
Therefore, the value of a₄ is 625.
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Todd is traveling to Mexico and needs to exchange $390 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?
find average speed that was traveled from city a to city p if trip took a half an hour to travel 23 miles
Step-by-step explanation:
Speed = distance / time
you are given distance = 23 miles and time = .5 hr
distance / time = 23 miles / .5 hr = 46 mph
Construct triangle PQR in which angle Q = 30 deg , angle R=60^ and PQ + QR + RP = 10cm
We can see here that in order to construct a triangle PQR in which angle Q = 30°, angle R=60° and PQ + QR + RP = 10cm, here is a guide:
Draw a line segment AB = 10 cm.Construct angle 30° at point A and angle 60° at point B.Draw angle bisectors to angles A and B.Make sure these angle bisectors intersect at point P.Draw perpendicular bisector to line segment AP.Let this bisector meet AB at Q.Then draw perpendicular bisector to line segment BP.Let this bisector meet AB at R.Join PQ and PR.PQR is the required triangle.What is a triangle?A triangle is a geometric shape that is defined as a three-sided polygon, where each side is a line segment connecting two of the vertices, or corners, of the triangle. The interior angles of a triangle always add up to 180 degrees.
Triangles can be classified into different types based on their side lengths and angles, such as equilateral triangles with three equal sides and three equal angles, isosceles triangles with two equal sides and two equal angles, and scalene triangles with no equal sides or angles.
Triangles are used in many areas of mathematics and science, including geometry, trigonometry, and physics.
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Find the closed formula for each of the following sequences by relating them to a well known sequence. Assume the first term given is a1.
(a) 2, 5, 10, 17, 26, . . .
(b) 0, 2, 5, 9, 14, 20, . . .
(c) 8, 12, 17, 23, 30, . . .
(d) 1, 5, 23, 119, 719, . . .
The final closed formula answers for each part,
(a) an = n^2 + 1
(b) an = n(n + 1)(n + 2)/6
(c) an = 2n + 6
(d) an = n! + (n-1)! + ... + 2! + 1!
(a) The given sequence can be seen as the sequence of partial sums of the sequence of odd numbers: 1, 3, 5, 7, 9, . . . . That is, the nth term of the given sequence is the sum of the first n odd numbers, which is n^2. Therefore, the closed formula for the given sequence is an = n^2 + 1.
(b) The given sequence can be seen as the sequence of partial sums of the sequence of triangular numbers: 1, 3, 6, 10, 15, . . . . That is, the nth term of the given sequence is the sum of the first n triangular numbers, which is n(n + 1)(n + 2)/6. Therefore, the closed formula for the given sequence is an = n(n + 1)(n + 2)/6.
(c) The given sequence can be seen as the sequence of differences between consecutive squares: 1, 5, 9, 16, 21, . . . . That is, the nth term of the given sequence is the difference between the (n+1)th square and the nth square, which is (n + 1)^2 - n^2 = 2n + 1. Therefore, the closed formula for the given sequence is an = 2n + 6.
(d) The given sequence can be seen as the sequence of partial sums of the sequence defined recursively by a1 = 1 and an+1 = an(n + 1) for n ≥ 1. That is, the nth term of the given sequence is the sum of the first n terms of the recursive sequence. It can be shown that the nth term of the recursive sequence is n! (n factorial), and therefore the nth term of the given sequence is the sum of the first n factorials. That is, an = 1 + 1! + 2! + ... + (n-1)! + n!. Therefore, the closed formula for the given sequence is an = n! + (n-1)! + ... + 2! + 1!.
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Use Synthetic Division to find the quotient of the division between
x4-2x3+2x2-9x +10 and (x-2).
x^3 + 2x - 5
x^3 + x + 4
x^4 + 2x - 5
x^2 - 2x + 5
Using synthetic division for the given dividend and divisor the required quotient is given by option a. x^3 + 2x - 5.
Dividend is equal to,
x^4-2x^3+2x^2-9x +10
Divisor is equal to,
( x - 2 )
Using synthetic division we have,
Use double equals to method to simplify the dividend we have,
x^4-2x^3+2x^2-9x +10
= x^4 - 2x^3 + 2x^2 - 4x -5x + 10
= x^3 (x -2) + 2x ( x -2 ) - 5 ( x - 2 )
= ( x - 2 ) ( x^3 + 2x - 5 )
Now ,
Simplify by dividing it,
( x^4-2x^3+2x^2-9x +10 ) ÷ ( x - 2 )
= ( x - 2 ) ( x^3 + 2x - 5 ) ÷ ( x -2 )
= ( x^3 + 2x - 5 )
Therefore, the quotient of the synthetic division for the given dividend is equal to option a . ( x^3 + 2x - 5 ).
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For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x) = 4x+9; g(x)=9x - 5
Answer:
(a) Find (f + g)(x)
To find (f + g)(x), we add the two functions f(x) and g(x):
(f + g)(x) = f(x) + g(x) = (4x + 9) + (9x - 5) = 13x + 4
The domain of (f + g)(x) is all real numbers, since there are no restrictions on x that would make (f + g)(x) undefined.
(b) Find (f - g)(x)
To find (f - g)(x), we subtract the function g(x) from f(x):
(f - g)(x) = f(x) - g(x) = (4x + 9) - (9x - 5) = -5x + 14
The domain of (f - g)(x) is all real numbers, since there are no restrictions on x that would make (f - g)(x) undefined.
(c) Find (f * g)(x)
To find (f * g)(x), we multiply the two functions f(x) and g(x):
(f * g)(x) = f(x) * g(x) = (4x + 9)(9x - 5) = 36x^2 + 11x - 45
The domain of (f * g)(x) is all real numbers, since there are no restrictions on x that would make (f * g)(x) undefined.
(d) Find (f / g)(x)
To find (f / g)(x), we divide the function f(x) by g(x):
(f / g)(x) = f(x) / g(x) = (4x + 9) / (9x - 5)
The domain of (f / g)(x) is all real numbers except x = 5/9, since this value would make the denominator of (f / g)(x) equal to zero, resulting in division by zero, which is undefined.
(e) Find f(g(x))
To find f(g(x)), we substitute g(x) into the expression for f(x):
f(g(x)) = 4g(x) + 9
Substituting the expression for g(x), we get:
f(g(x)) = 4(9x - 5) + 9 = 36x - 11
The domain of f(g(x)) is all real numbers, since there are no restrictions on x that would make f(g(x)) undefined.
(f) Find g(f(x))
To find g(f(x)), we substitute f(x) into the expression for g(x):
g(f(x)) = 9f(x) - 5
Substituting the expression for f(x), we get:
g(f(x)) = 9(4x + 9) - 5 = 36x + 76
The domain of g(f(x)) is all real numbers, since there are no restrictions on x that would make g(f(x)) undefined.
(g) Find f(f(x))
To find f(f(x)), we substitute f(x) into the expression for f(x):
f(f(x)) = 4f(x) + 9
Substituting the expression for f(x), we get:
f(f(x)) = 4(4x + 9) + 9 = 16x + 45
The domain of f(f(x)) is all real numbers, since there are no restrictions on x that would make f(f(x)) undefined.
(h) Find g(g(x))
To find g(g(x)), we substitute g(x) into the expression for g(x):
g(g(x)) = 9
Step-by-step explanation:
Answer:
Step-by-step explanation:
(a) Find f(g(x)).
To find f(g(x)), we first need to find g(x) and then substitute it into f(x).
g(x) = 9x - 5
f(g(x)) = f(9x - 5) = 4(9x - 5) + 9 = 36x - 11
Therefore, f(g(x)) = 36x - 11.
(b) Find g(f(x)).
To find g(f(x)), we first need to find f(x) and then substitute it into g(x).
f(x) = 4x + 9
g(f(x)) = g(4x + 9) = 9(4x + 9) - 5 = 36x + 76
Therefore, g(f(x)) = 36x + 76.
(c) Find f(f(x)).
To find f(f(x)), we need to substitute f(x) into f(x).
f(f(x)) = 4(4x + 9) + 9 = 16x + 45
Therefore, f(f(x)) = 16x + 45.
(d) Find g(g(x)).
To find g(g(x)), we need to substitute g(x) into g(x).
g(g(x)) = 9(9x - 5) - 5 = 81x - 50
Therefore, g(g(x)) = 81x - 50.
Domain of f(x) and g(x): Since both f(x) and g(x) are linear functions, their domains are all real numbers.
(e) Find the inverse of f(x).
To find the inverse of f(x), we need to switch the roles of x and f(x) and solve for f(x).
y = 4x + 9
x = 4y + 9
x - 9 = 4y
y = (x - 9) / 4
Therefore, the inverse of f(x) is f^(-1)(x) = (x - 9) / 4.
(f) Find the inverse of g(x).
To find the inverse of g(x), we need to switch the roles of x and g(x) and solve for g(x).
y = 9x - 5
x = 9y - 5
x + 5 = 9y
y = (x + 5) / 9
Therefore, the inverse of g(x) is g^(-1)(x) = (x + 5) / 9.
(g) Find the domain of f^(-1)(x).
The domain of f^(-1)(x) is the range of f(x). Since f(x) is a linear function, its range is all real numbers. Therefore, the domain of f^(-1)(x) is also all real numbers.
(h) Find the domain of g^(-1)(x).
The domain of g^(-1)(x) is the range of g(x). Since g(x) is a linear function, its range is all real numbers. Therefore, the domain of g^(-1)(x) is also all real numbers.
For the functions f(x)=−7x+3 and g(x)=3x2−4x−1, find (f⋅g)(x) and (f⋅g)(1).
Answer:
Find f(g(x)) f(x)=7x-8 , g(x)=3x-2. f(x)=7x−8 f ( x ) = 7 x - 8 , g(x)=3x−2 g ( x ) = 3 x - 2. Step 1. Set up the composite result function. f(g(x)) f ( g ...
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The dot plots below show the number of students in attendance each day in Mr. Wilson's class and Mr. Watson's class in April. What is the difference of the medians as a multiple of the interquartile range? A. B. C. D.
The difference of the medians as a multiple of the interquartile range is 0.5,So the correct answer is option (A) 0.5.
What is median?The median is a measure of central tendency that represents the middle value in a data set when the values are arranged in numerical order.
For example, consider the data set {3, 5, 2, 6, 1, 4}. When the values are ordered from smallest to largest, we get {1, 2, 3, 4, 5, 6}. The median in this case is the middle value, which is 3.
We can first find the medians and interquartile ranges of the two dot plots.
For Mr. Wilson's class:
Median = 12
Q1 = 10
Q3 = 14
IQR = Q3 - Q1 = 14 - 10 = 4
For Mr. Watson's class:
Median = 10
Q1 = 8
Q3 = 12
IQR = Q3 - Q1 = 12 - 8 = 4
The difference of the medians is |12 - 10| = 2. Therefore, the difference of the medians as a multiple of the interquartile range is:
$$\frac{2}{4} = \boxed{0.5}$$
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Help i need this question solved
Answer: D. Square
Step-by-step explanation:
The shape created by the cross section of the cut through a square pyramid is a square.
To see why, imagine the pyramid sitting on a table with its square base flat against the surface. The cut goes through the vertex, which is the point at the top of the pyramid. Since the cut is perpendicular to the base, it divides the pyramid into two smaller pyramids with congruent, but not identical, bases. Each of these smaller pyramids has a triangular base that is an isosceles right triangle. The two triangles share a common hypotenuse, which is the line of the cut.
The cross section of the cut is the shape formed where the two triangles meet along the hypotenuse. Since both triangles are congruent and the hypotenuse is the same for both, the cross section is a square. The sides of the square are equal to the base of the original pyramid, which is one of the legs of the isosceles right triangles formed by the cut. Therefore, the answer is D, a square.
Which expressions are equivalent to 8(3/4y - 2) + 6(-1/2x + 4) + 1
Answer:
8(3/4y - 2) + 6(-1/2x + 4) + 1 can be simplified as:
6(-1/2x) = -3x
8(3/4y) = 6y
8(-2) = -16
6(4) = 24
1 remains as 1.
So the expression becomes:
6y - 3x - 16 + 24 + 1
which simplifies to:
6y - 3x + 9
Therefore, the expressions that are equivalent to 8(3/4y - 2) + 6(-1/2x + 4) + 1 are:
6y - 3x + 9
