hi! i need help checking these problems!!
1) 12x + 9 = -15 i know the answer i just need help checking!
2) x/7 - 4 = 4 same with this i need help checking it!

Required true statements are B, C, D, G, H, I, L, M.

What is factor?

In mathematics, factors refer to the numbers that can be multiplied together to obtain a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Based on the given factors, we can conclude that:

f(x) has a factor of 3, which means that 3 is a root/solution of f(x).

f(x) has a factor of (2x-1), which means that 2x-1=0 or x=1/2 is a root/solution of f(x).

f(x) has a factor of (x+5), which means that x+5=0 or x=-5 is a root/solution of f(x).

Using these facts, we can determine which statements are true:

A) -1/2 is not a root/solution of f(x). (False)

B) We can check if this expression is equivalent to f(x) by factoring it:

6x² - 33x - 15 = 3(2x² - 11x - 5) = 3(x - 5)(2x + 1).

This shows that f(x) has factors of 3, (2x-1) and (x+5), so this statement is true.

C) -5 is a root/solution of f(x). (True)

D) 3 is a root/solution of f(x). (True)

E) We can check if this expression is equivalent to f(x) by factoring it:

2x²+9x-5 = (2x - 1)(x + 5), which is not the same as the factors given for f(x), so this statement is false.

F) 0 is not necessarily a root/solution of f(x), since the factors given do not guarantee that f(x) intersects the x-axis at 0. (False)

G) We can check if this expression is equivalent to f(x) by factoring it:

6x²+27x-15 = 3(2x²+9x-5) = 3(2x-1)(x+5), which shows that f(x) has the same factors as this expression, so this statement is true.

H) 1/2 is a root/solution of f(x). (True)

I) 5 is a root/solution of f(x). (True)

J) We can check if this expression is equivalent to f(x) by factoring it:

2x²-9x-5 = (2x + 1)(x - 5), which is not the same as the factors given for f(x), so this statement is false.

K) -3 is not necessarily a root/solution of f(x), since the factors given do not guarantee that f(x) intersects the x-axis at -3. (False)

L) To find the x-intercepts of f(x), we need to set f(x) equal to zero and solve for x:

f(x) = 3(x + 5)(2x - 1) = 0

This equation has roots of x=-5 and x=1/2, which means that the x-intercepts of f(x) are (-5, 0) and (1/2, 0). So, this statement is true.

M) To find the x-intercepts of f(x), we need to set f(x) equal to zero and solve for x:

f(x) = 3(x + 5)(2x - 1) = 0

This equation has roots of x=-5 and x=1/2, which means that the x-intercepts of f(x) are (-5, 0) and (1/2, 0).

So, this statement is true.

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The correct area models are 4e and none of the given models for 4d and 4c.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The area model of 4c is incorrect because the last term of the polynomial, which is 40, is represented by the bottom right box of the area model. However, in the given area model, the box representing 40 is in the upper row.

The area model of 4e is correct because it represents the polynomial as the product of two binomials, where the factors are (x-5) and (x-8).

When multiplied using the distributive property, the resulting polynomial is x² - 13x + 40, which matches the given polynomial form.

The area model of 4d is incorrect because it does not properly represent a factorable polynomial.

It only includes three boxes, representing the three terms of the polynomial, but it does not show how the terms can be factored into binomials.

Therefore, the correct area models are 4e and none of the given models for 4d and 4c.

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The probability that the chosen integer is a perfect square or a perfect cube but not both is therefore 13/100.

When choosing an integer from 1 to 100, there are 10 perfect squares (1² to 10²) and 4 perfect cubes (1³ to 4³). However, the number 1 is both a perfect square and a perfect cube. To avoid counting it twice, we use the formula for the union of two sets: |A∪B| = |A| + |B| - |A∩B|. In this case, |A∪B| represents the number of integers that are either a perfect square or a perfect cube. Plugging in the values, we get |A∪B| = 10 + 4 - 1 = 13. The probability is therefore 13/100.

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Answer:151

Step-by-step explanation:

Addition of 107 and 44

The value of the angle m∠EFG is 151.

We have,

From  the figure,

m∠EFG = m∠EFH + m∠HFG

Now,

Substituting the values in the angle.

m∠EFG

= m∠EFH + m∠HFG

= 44 + 107

= 151

Thus,

The value of the angle m∠EFG is 151.

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The total surface area of the tower, including the bottom base, is [tex]3600+400\sqrt{2}[/tex] square meters.

The surface area of the tower can be found by calculating the surface area of the prism and the surface area of the pyramid separately, and then adding them together.

The surface area of the square base of the prism is [tex]20^2 = 400[/tex] square meters.

The lateral surface area of the prism is the product of the perimeter of the base and the height, which is [tex]4*(20)*40 = 3200[/tex] square meters.

The surface area of the pyramid is given by the formula:

base area + 1/2(perimeter of base)(slant height)

The base area of the pyramid is [tex]20^2 = 400[/tex] square meters. The perimeter of the base is 4(20) = 80 meters. Therefore, the surface area of the pyramid is:

[tex]400 + [(\frac{1}{2})*(80)(10\sqrt{2})] = 400 + 400\sqrt{2}[/tex] square meters.

The total surface area of the tower is the sum of the surface area of the prism and the surface area of the pyramid, which is:

[tex]3200 + 400 + 400\sqrt{2} = 3600 + 400\sqrt{2}[/tex] square meters.

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Answer:

2035.752 in³/min

Step-by-step explanation:

To find the rate of change of the volume of the sphere at the instant its radius is 9 inches, we need to work out dV/dt when r = 9.

The equation for the volume of a sphere is:

[tex]V=\dfrac{4}{3}\pi r^3[/tex]

Differentiate the expression for volume with respect to r:

[tex]\begin{aligned}V&=\dfrac{4}{3}\pi r^3\\\\\implies\dfrac{\text{d}V}{\text{d}r}&=3 \cdot \dfrac{4}{3} \pi r^{3-1}\\\\\dfrac{\text{d}V}{\text{d}r}&=4 \pi r^2\end{aligned}[/tex]

We know that that the radius of a sphere is increasing at a constant rate of 2 inches per minute, so the rate of change of the radius of sphere is:

[tex]\dfrac{\text{d}r}{\text{d}t}=2[/tex]

To find an expression for dV/dt, use the chain rule:

[tex]\boxed{\dfrac{\text{d}V}{\text{d}t}=\dfrac{\text{d}V}{\text{d}r} \times \dfrac{\text{d}r}{\text{d}t}}[/tex]

Substitute the expressions for dV/dr and dr/dt to create an expression for dV/dt:

[tex]\begin{aligned}\implies \dfrac{\text{d}V}{\text{d}t}&=4 \pi r^2 \times 2\\&=8 \pi r^2\end{aligned}[/tex]

To find the value of dV/dt when r = 9, substitute r = 9 into the equation for dV/dt:

[tex]\begin{aligned}\dfrac{\text{d}V}{\text{d}t}\;\textsf{at}\;r=9\implies \dfrac{\text{d}V}{\text{d}t}&=8 \pi (9)^2\\&=8\pi (81)\\&=648\pi\\&=2035.752\; \sf in^3/min\;(3\;d.p.)\end{aligned}[/tex]

Therefore, the rate of change of the volume of the sphere at the instant when its radius is 9 inches is 2035.752 in³/min (rounded to three decimal places).

Answer: The angle elevation is 34.6 9 degree.

Step-by-step explanation:

Brainliest pls:)

Answer:

3 and 1/2

Step-by-step explanation:

7 and 1/2 - 4

The equation y = 2x + 15, on the other hand, does not illustrate a proportional connection since it cannot be stated in the form y = kx. As x grows, y increases by a factor of two.

In mathematics, an equation is a statement that states the equivalence of two expressions. An equation is made up of two sides separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" contends that the phrase "2x Plus 3" equals the number "9." The purpose of equation solving is to identify the value or values of the variable(s) that will make the equation true. Simple or complicated equations, regular or nonlinear, with one or more components are all possible. For example, in the equation "x2 + 2x - 3 = 0," the variable x is raised to the second power. Lines are utilized in many areas of mathematics, including algebra, calculus, and geometry.

A. There is no one optimum approach to determine if an equation indicates a proportionate connection. One popular method is to test if the equation can be stated in the form y = kx, where k is a constant. A proportional connection is represented by an equation that may be expressed in this format.

A. Because it can be expressed as y = kx with k = 2, the equation y = 2x represents a proportional connection. As a result, D is the right answer: just y = 2x.

The equation y = 2x + 15, on the other hand, does not illustrate a proportional connection since it cannot be stated in the form y = kx. As x grows, y increases by a factor of two.

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The given statement "Using p-value rule for a population proportion or mean, if the level of significance is less than p-value, null hypothesis is rejected." is True because the hypothesis is rejected in this case.

In hypothesis testing, the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, assuming that the null hypothesis is true. The level of significance, denoted by alpha, is the maximum probability of rejecting the null hypothesis when it is actually true.

If the p-value is less than the level of significance, it means that the observed test statistic is unlikely to have occurred by chance alone, assuming the null hypothesis is true. Therefore, we reject the null hypothesis in favor of the alternative hypothesis at the given level of significance.

For example, suppose we are testing the hypothesis that the population mean is equal to a certain value. If the p-value is 0.02 and the level of significance is 0.05, we would reject the null hypothesis because the p-value is less than the level of significance.

This means that there is strong evidence against the null hypothesis and we can conclude that the population mean is likely different from the hypothesized value.

In summary, if the level of significance is less than the p-value, we reject the null hypothesis in favor of the alternative hypothesis at the given level of significance.

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The value of k given the system of equations is 2

Given that

x + y + 3z = 10

4x + 2y + 5z = 7

kx + z = 3

Eliminate y in the first two equations

So, we have

2x + 2y - 6z = 20

4x + 2y + 5z = 7

kx + z = 3

Subtract the first two equations

So, we have

2x + z = -13

kx + z = 3

When it has no solution, we have

kx + z = 2x + z

Evaluate

k = 2

Hence, the value of k is 2

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If  x is the unknown number and Nineteen more than a number. Thus, Translation is 19 units on the right to get x + 19.

Each point in a figure is moved the same distance in a single direction using a type of transformation known as translation.

In geometry, a function that transfers an object a specific distance is referred to as translation. Nothing else is done to change the object. It is not scaled, reflected, rotated, or refracted.

Give that:

x is the unknown number and Nineteen more than a number.

Then, there is a shift of 19 units on the right of x. add 19 to unknown number x to get the translation.

x + 19

Thus, Translation is 19 units on the right to get x + 19.

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Answer:

A)  The dimensions of the new triangular deck are:

B)  Area of original deck = 63 square feet
     Area of new deck = 393.75 square feet

C)  The ratio of the area is 4 : 25. This is the square of the scale factor.

Step-by-step explanation:

The given scale is 2 : 5. This means that the ratio of the measurements of the corresponding sides of two objects is 2 to 5. In other words, if one object has a length of 2 units, the corresponding length of the other object is 5 units.  So in this scenario, a 2 : 5 scaled version means that for every 2 foot of the original deck, there is 5 foot of the new deck.

Therefore, if the original base of the triangle is 18 feet, the new base, b, will be:

[tex]\implies \sf 2 : 5 = 18 : b[/tex]

[tex]\implies \sf \dfrac{2}{5} = \dfrac{18}{b}[/tex]

[tex]\implies \sf 2 \cdot b=18 \cdot 5[/tex]

[tex]\implies \sf b=\dfrac{18 \cdot 5}{2}[/tex]

[tex]\implies \sf b=45\;ft[/tex]

Similarly, if the original height of the triangle is 7 feet, the new height, h, will be:

[tex]\implies \sf 2 : 5 = 7 : h[/tex]

[tex]\implies \sf \dfrac{2}{5} = \dfrac{7}{h}[/tex]

[tex]\implies \sf 2 \cdot h=7 \cdot 5[/tex]

[tex]\implies \sf h=\dfrac{7\cdot 5}{2}[/tex]

[tex]\implies \sf h=17.5\;ft[/tex]

Therefore, the dimensions of the new triangular deck are:

[tex]\hrulefill[/tex]

The area of a triangle is half of the product of its base and height:

[tex]\boxed{\sf A=\dfrac{1}{2}bh}[/tex]

Area of the original deck:

[tex]\implies \sf A=\dfrac{1}{2} \cdot 18 \cdot 7[/tex]

[tex]\implies \sf A=9 \cdot 7[/tex]

[tex]\implies \sf A=63\;ft^2[/tex]

Area of the new deck:

[tex]\implies \sf A=\dfrac{1}{2} \cdot 45\cdot 17.5[/tex]

[tex]\implies \sf A=22.5 \cdot 17.5[/tex]

[tex]\implies \sf A=393.75\;ft^2[/tex]

Therefore, the areas of the two decks are:

[tex]\hrulefill[/tex]

The ratio of the area of the original deck to the area of the new deck is:

[tex]\implies \textsf{Area original deck}:\textsf{Area new deck}=\sf 63 : 393.75[/tex]

To rewrite the ratio in its simplest form, multiply both sides of the ratio by 16:

[tex]\implies \sf 63 \cdot 16: 393.75 \cdot 16=1008 :6300[/tex]

Then divide both sides of the ratio by 252:

[tex]\implies \sf \dfrac{1008}{252}:\dfrac{6300}{252}=4:25[/tex]

Therefore, the ratio of the area of the original deck to the area of the new deck in its simplest terms is 4 : 25.

If we compare this to the original scale factor 2 : 5 (which is the ratio of length), we can see that the ratio of area is the square of the scale factor:

[tex]\implies \sf 2^2:5^2=4:25[/tex]

The measure of angle 14 is 83° in the parallel line t and s and transversal n.

Parallel lines are a pair of lines in a two-dimensional plane that never intersect or cross each other, regardless of how far they are extended in either direction. Parallel lines have the same slope, which determines their steepness or inclination, and they maintain the same slope throughout their length.

A transversal is a line that intersects two or more other lines in a plane at distinct points. Specifically, if two or more lines in a plane are intersected by a third line, that third line is called a transversal. The lines being intersected are referred to as the "transversed lines".

According to the given information:

Given the parallel lines are t and s, the transversal are line m and n

the given angles are ∠2 = 97° and ∠6 = 83°

In the parallel lines t and s , line n is transversal

By the property that "corresponding angles are equal"

we can say that ∠5 =∠13, ∠7 =∠15, ∠6= ∠14, ∠8=∠16

given ∠6 = 83°, We know that ∠6= ∠14

∠14 = 83°       (Corresponding angles are equal)

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Answer:

Step-by-step explanation:

Refers to the attachment given below!

By answering the presented question, we may conclude that equation Asymptotes: y = ±(3/9)x and y = ±(-3/9)x, which simplify to y = ±(1/3)x and y = ∓(1/3)x. and The hyperbola opens Vertically.

In mathematics, an equation is a statement that states the equivalence of two expressions. An equation is made up of two sides separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" contends that the phrase "2x + 3" equals the number "9." The purpose of equation solving is to identify the value or values of the variable(s) that will make the equation true. Simple or complicated equations, regular or nonlinear, with one or more components are all possible. For example, in the equation[tex]"x2 + 2x - 3 = 0,[/tex]" the variable x is raised to the second power. Lines are utilized in many areas of mathematics, including algebra, calculus, and geometry.

Center: (0,0)

Transverse axis length: 2a = 2*3 = 6

Conjugate axis length: 2b = 2*9 = 18

Vertices: (0,±3)

Foci: (0,±√18)

Asymptotes: y = ±(3/9)x and y = ±(-3/9)x, which simplify to y = ±(1/3)x and y = ∓(1/3)x.

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Answer:

Step-by-step explanation:

The formula to calculate the volume of a cone is:

v = h * π * r ^ 2/3

Where,

h: height

r: radio

We then clear the value of h:

h = (3 * v) / (π * r ^ 2)

We substitute the values:

h = (3 * (118π)) / (π * ((2/3) / (2)) ^ 2)

h = 3186 ft

Answer:

The height of the cone is:

h = 3186 ft

The probability that owners ranked Subaru third is 2/6, which reduces to 1/3. Therefore, the correct answer is 1/3.

To find the probability that owners ranked Subaru third, we need to find out the total number of possible rankings for

the three dealers: Subaru, Honda, and Buick.

There are three possible rankings for the first choice, two for the second choice, and one for the third choice.

Therefore, there are 3 x 2 x 1 = 6 possible rankings for the dealers.

Since each of the possible rankings is equally likely, the probability that owners ranked Subaru third is the number of

ways to rank Subaru third divided by the total number of possible rankings.

Subaru can be ranked third in two out of the six possible rankings:

Subaru, Honda, Buick Honda, Buick, Subaru

Therefore, the probability that owners ranked Subaru third is 2/6, which reduces to 1/3.

Therefore, the correct answer is 1/3.

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The slope of a line segment represent in the context of this situation is the total wages for the week.

We have seen lines drawn on the coordinate plane in geometry. The best approach to determine without using any geometrical tools if the lines are parallel, perpendicular, or at any angle is to measure the slope.

A line's slope in mathematics is defined as the ratio of the change in the y coordinate to the change in the x coordinate.

Both the net change in the y-coordinate and the net change in the

x-coordinate are denoted by y and x, respectively.

Thus, the formula for the change in y-coordinate with regard to the change in x-coordinate is

[tex]m=\frac{y}{x} =\frac{change \ in \ y}{change\ in\ x}[/tex]

You are a crew at a Convenience store that pays an hourly wage $ 7 and 1.5 times the hours wage for the extra hours if you work for more than 40 hours a week. Write a piecewise function that gives the weekly pay P in term of the number of hours h your work.

→ wages for 1 hours = $ 7

So,

we can say that, wages for 40 hours, = 7 x 40 = 280 .

Therefore,

→ Total wages of week is = 280.

Piecewise function is :-

P(h) = 280 , where h ≤ 40.

Now, we have given that, store gives 1.5 times the hours wage for the extra hours if you work for more than

40 hours a week.

So,

→ wages per hour now = 1.5 x 7 = 10.5 .

Therefore,

→ Total wages of week

= 280 + 10.5(h - 40)

= 280 + 10.5h - 420

= 10.5h - 140.

and,

Piecewise function that gives the weekly pay P in term of the number of hours h your work is:

P(h) = 280 + 10.5(h - 40), where [h > 40].

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115.

The point estimate for the average IQ score of the population of 3000 college students based on a simple random sample of 150 students with an average IQ score of 115 and a standard deviation of 10 is 115.

Therefore, the correct option is 115.

A point estimate is an estimate of a population parameter based on a single value or point. It is obtained by using statistics collected from a sample to infer population values. It can be a single value or a range of values that best represents the unknown population parameter.

Based on the given information, a simple random sample of 150 students is drawn from a population of 3000 college students.

Among sampled students, the average IQ score is 115 with a standard deviation of 10. The point estimate, in this case, is the average IQ score of the population, which is estimated to be 115 based on the sample data.

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a. The probability that the amount of collagen is greater than 60 grams per mililiter is 0.5398

b. The probability that the amount of collagen is less than 90 grams per mililiter is 0.9985.

c. Approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

a. To find the probability that the amount of collagen is greater than 60 grams per milliliter, we'll use the z-score formula:
z = (X - μ) / σ
where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
z = (60 - 61) / 9.8 ≈ -0.1
Now, we'll use a z-table or calculator to find the probability corresponding to this z-score.

A z-score of -0.1 corresponds to a probability of approximately 0.4602.

Since we're looking for the probability that the amount of collagen is greater than 60 grams per milliliter, we need to find the area to the right of this z-score:
P(X > 60) = 1 - P(X ≤ 60) = 1 - 0.4602 ≈ 0.5398
b. To find the probability that the amount of collagen is less than 90 grams per milliliter, we'll again use the z-score formula:
z = (90 - 61) / 9.8 ≈ 2.96
A z-score of 2.96 corresponds to a probability of approximately 0.9985.

Since we're looking for the probability that the amount of collagen is less than 90 grams per milliliter, we'll use this probability directly:
P(X < 90) ≈ 0.9985
c. To find the percentage of compounds formed from the extract of this plant that fall within 1 standard deviation of the mean, we'll use the empirical rule (68-95-99.7 rule), which states that approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

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The product of equation (-a + 3)(a + 4) is -a² - a + 12.

What is product of equation?

Using the distributive property, we can expand the product of (-a + 3)(a + 4) as follows:

(-a + 3)(a + 4) = -a(a) - a(4) + 3(a) + 3(4)

Simplifying, we get:

(-a + 3)(a + 4) = -a² - 4a + 3a + 12

Combining like terms, we get:

(-a + 3)(a + 4) = -a² - a + 12

Therefore, the product of (-a + 3)(a + 4) is -a² - a + 12.

What is distributive property?

The distributive property is a property of arithmetic operations that is used to simplify expressions by distributing one term over another term inside parentheses.

In particular, the distributive property states that:

a × (b + c) = (a × b) + (a × c)

or

(a + b) × c = (a × c) + (b × c)

This property applies to both multiplication and addition, and can be extended to subtraction and division by adding or subtracting the distributive term to both sides of the equation.

For example, using the distributive property, we can simplify the expression:

3 × (4 + 5)

as follows:

3 × (4 + 5) = 3 × 4 + 3 × 5

= 12 + 15

= 27

In this example, we used the distributive property to distribute the term 3 over the sum (4 + 5), resulting in the equivalent expression 3 × 4 + 3 × 5.

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Therefore , the solution of the given problem of average comes out to be  96% confidence interval is ($52,791.38, $53,086.62).

The result of a collection, also known as the arithmetic mean, is the sum of all values split by all of the values. One of the most commonly used primary trend indicators, it is frequently referred to as "mean." Multiply the total number of numbers in the collection by each value to get the result. Calculations can be performed using either the raw data or data that has been merged into frequency charts.

Here,

The error margin (E) is equal to z* (standard deviation / sqrt(n)).

Interval of confidence equals sample mean  E

Where n is the sample size, z* is the critical value for the required confidence level, and s is the population standard deviation.

We can enter the provided values into the margin of error formula as follows:

=> E = 2.05 * (1227 / sqrt(227))

=>  E ≈ 147.62

Therefore, the error range is roughly $147.62.

Next, we can create the confidence interval using the group mean and margin of error:

Interval of confidence equals sample mean  E

=>  Confidence range is between $52,939 and $147.62.

=>  Interval of confidence = ($52,791.38, $53,086.62)

Therefore, the range of the typical household income in Martin County within the 96% confidence interval is ($52,791.38, $53,086.62).

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Answer:

The new value of the list after changing 71° to 93° would be:

58, 61, 93, 77, 91, 100, 105, 102, 95, 82, 66, 57

The measure that changes the most would be the median. Without the change, the median was 86.5°, which was the average of the 7th and 8th values in the list (100 + 77) / 2. However, after changing 71° to 93°, the median becomes 91°, which is the average of the 6th and 7th values in the list (91 + 91) / 2.

Therefore, the median has changed by 4.5° (from 86.5° to 91°).

Hue is arranging chair, She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. There are 7 chairs in that row length.

A row is anything that is arranged in a series, such as a sequence of figures on a worksheet or stitches on a knitting needle. A row is anything that is arranged in a straight line, such as a line of penguins at the zoo, tulips in a yard, or tuba players moving in your town's Fourth of July procession.

Suppose the number of chairs in a row is 'x' and total number of chairs are 'y'.

Given:

Hue can form 6 rows of a given length with 3 chairs left over.

From that we get

y=6x+3 ………… (1)

Or she can form 8 rows of that same length if she gets 11 more chairs.

We get

y=8x-11 …………(2)

We can put the value of y from the equation 1 in equation 2

6x+3=8x-11

Or, 2x=14

Or, x=7

X is the number of chairs present in a row length. So that is 7.

Hence there are 7 chairs in that row length

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Based on the IQR values, we conclude that Bus 18 is more consistent than Bus 47, as it has smaller IQR. This means the travel times of students on Bus 18 are more tightly clustered around the median, and are fewer outliers in data.

A line plot is a type of graph that displays data as points or dots along a number line. Each dot or point represents a single data value, and the number line shows the range of the data. Line plots are useful for displaying small to moderate-sized data sets and for identifying the central tendency and variability of the data.

To determine which bus is the most consistent, we need to look at the variability of data. The most appropriate measure of variability for this purpose is the interquartile range (IQR), which is the range of the middle 50% of the data.

From the given data, we can calculate the IQR for both buses as follows:

For Bus 18:

Lower quartile (Q1) = 8

Upper quartile (Q3) = 24

IQR = Q3 - Q1 = 24 - 8 = 16

For Bus 47:

Lower quartile (Q1) = 10

Upper quartile (Q3) = 34

IQR = Q3 - Q1 = 34 - 10 = 24

The range, which is the difference between the maximum and minimum values, is not as appropriate a measure of variability for this purpose, as it is heavily influenced by outliers. Therefore, we cannot use the range to determine the consistency of the data for either bus.

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We should choose a sample size of at least 153 to be 95% confident that the true average waiting time is within 7 days of the sample mean.

In statistics, a confidence interval is a range of values that is used to estimate an unknown population parameter with a certain degree of confidence or probability. It is a statistical measure that indicates the level of uncertainty or variation associated with a particular estimate or sample.

A confidence interval is typically expressed as a range of values, along with a level of confidence, such as 95% or 99%. This means that if the same study were to be repeated many times, the calculated confidence interval would contain the true population parameter for the given confidence level in a specified proportion of the repetitions. For example, a 95% confidence interval means that if the same study were repeated many times, the interval would contain the true population parameter 95% of the time.

We can use the formula for the margin of error of a confidence interval to solve this problem:

Margin of error = z*(sigma/√(n))

where:

z = the z-score associated with the desired level of confidence (in this case, 1.96 for 95% confidence)

sigma = the population standard deviation (15 days)

n = the sample size we want to find

We want the margin of error to be no more than 7 days, so we can set up the following inequality:

7 <= 1.96*(15/√(n))

Solving for n, we get:

n >= (1.96*15/7)²

n >= 12.36²

n >= 152.7

Therefore, we should choose a sample size of at least 153 to be 95% confident that the true average waiting time is within 7 days of the sample mean.

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With D as Center of dilation, the scale factor is √2.

In mathematics, a scaling factor is a ratio that describes how much a figure or object has been enlarged or reduced in size. A scaling factor is typically expressed as a fraction or decimal that represents the relative change in size between the original figure and the new, scaled figure.

Now,

To find the scale factor of dilation, we can use the distance formula to find the ratio of corresponding sides in the original and dilated triangles.

The distance between points A and B in the original triangle is:

[tex]\rm AB = \sqrt{(x_2 - x_1)\² + (y_2 - y_1)\²}[/tex]

[tex]= \sqrt{(0 - (-1))\² + (1 - (-1))\²} \\\\= \sqrt{1\² + 2\²}[/tex]

= √5

The distance between points A' and B' in the dilated triangle is:

[tex]\rm A'B' = \sqrt{(x_2' - x_1')\² + (y_2' - y_1')\²}[/tex]

[tex]= \sqrt{(-2 - (-1))\² + (-5 - (-2))\²}[/tex]

[tex]= \sqrt{1\² + 3\²}[/tex]

= √10

Therefore, the scale factor for the dilation is:

Scale factor = A'B' / AB = √10 / √5 = √2

Hence, the scale factor of dilation is √2.

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Hence, a cylinder with height h and base area [tex]\pi r^{2}[/tex] may be proven to have the same volume as a triangular prism.

Two parallel circular bases joined by a curving surface form the three-dimensional geometric object known as a cylinder. Due of its parallel and congruent bases, it is a sort of prism.

When someone uses the word "cylinder,” they often mean a right circular cylinder with circles for bases and an axis that is perpendicular to the bases' planes.

A cylinder's volume may be calculated using the formula V = [tex]r^{2}[/tex]h, where r denotes the perimeter of the base and h the height of the cylinder. L = 2rh, where r is the radius of a base and h is the height of the cylinder, is the formula for a cylinder's lateral surface area.

Cavalieri's principle states that if two solid objects have the same height and every cross-section taken perpendicular to a common axis has the same area, then the two objects have the same volume.

Let's consider the given triangular prism with base area  [tex]\pi[/tex][tex]r^2[/tex] and height h. If we take a cross-section perpendicular to the base at a height y, we get a circle with radius r multiplied by a triangle with base 2r and height y. The area of this cross-section is:

[tex]A(y) = \pi r^2 + (2r)(y) / 2[/tex]

Simplifying, we get:

[tex]A(y) = \pi r^2 + ry[/tex]

Now, let's consider a cylinder with base area pi r² and height h. If we take a cross-section perpendicular to the height at a height y, we get a circle with radius r multiplied by a rectangle with base 2r and height h. The area of this cross-section is:

[tex]A'(y) = \pi r^ + (2r)(h) = \pi r^2 + 2rh[/tex]

By Cavalieri's principle, the triangular prism and the cylinder have the same volume if A(y) = A'(y) for all y from 0 to h. Let's check:

[tex]\pi r^² + ry = \pi r^² + 2rh[/tex]

[tex]ry = 2rh[/tex]

[tex]y = 2r[/tex]

So, we see that the areas are equal for all y between 0 and h, except at [tex]y = 2r.[/tex] However, this is just a single point and has no effect on the volume,

Therefore,  the cylinder with base area [tex]\pi[/tex][tex]r²[/tex] and height h has the same volume as the triangular prism, with base area [tex]\pi[/tex]r² and height h.

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Answer:

Step-by-step explanation:

This can be solved using Pythagorean Theorem.

a² + b² = c²

8383²  +  b²   = 158158²

70274689 + b² = 25013952964

70274689 - 70274689 + b² = 25013952964 - 70274689

    b² = 24943678275

√b²  = √24943678275

b = 157935.7 cm