If two triangles are congruent, which of the following statements must be true? Check all that apply . A. The triangles have the same shape. B. The triangles have the same size. C. The corresponding angles of the triangles are congruent. D. The corresponding sides of the triangles are congruent.

The extreme values of f(x, y) = xy occur at the points (2, 1) and (-2, -1) on the ellipse [tex]3x^{2} +2y^{2} =1[/tex].

To find the extreme values of f(x, y) = xy on the ellipse [tex]3x^{2} +2y^{2} =1[/tex], we can use the method of Lagrange multipliers.

Define the function g(x, y) = [tex]3x^{2} +2y^{2} -1[/tex]. We need to find points (x, y) where the gradient of f is proportional to the gradient of g:

∇f = λ∇g

The gradient of f is ∇f = (y, x), and the gradient of g is ∇g = (6x, 4y). Therefore, we have the following system of equations:

y = 6λx
x = 4λy

Substitute the second equation into the first:

y = 6λ(4λy)
y = [tex]24λ^{2y}[/tex]

If y ≠ 0, then 1 = [tex]24λ^{2}[/tex], and λ = ±1/2. Plugging this value into the second equation gives x = ±2. Thus, we have two potential extreme points: (2, 1) and (-2, -1).

Now consider the case when y = 0. The constraint equation becomes [tex]3x^{2} =1[/tex], and x = ±1/√3. However, these points correspond to f(x, y) = 0, which is not an extreme value.

Therefore, the extreme values of f(x, y) = xy occur at the points (2, 1) and (-2, -1) on the ellipse [tex]3x^{2} +2y^{2} =1[/tex].

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To create an array of 6 shape references named "shapes" and fill it with an assortment of shape, circle and square objects, 1) Define the base `Shape` class along with derived classes `Circle` and `Square`. 2) Create an array of `Shape` references named `shapes` with a size of 6. 3) Instantiate various `Circle` and `Square` objects and assign them to the elements in the `shapes` array.

To create an array of 6 shape references named "shapes" and fill it with an assortment of shape, circle and square objects, you can use the following code in a programming language like Java:
Shape[] shapes = new Shape[6]; // create an array of 6 shape references
shapes[0] = new Shape(); // add a generic shape object to the array
shapes[1] = new Circle(); // add a circle object to the array
shapes[2] = new Square(); // add a square object to the array
shapes[3] = new Circle(); // add another circle object to the array
shapes[4] = new Square(); // add another square object to the array
shapes[5] = new Circle(); // add a third circle object to the array
This code first declares an array of 6 shape references named "shapes" using the syntax `Shape[] shapes = new Shape[6];`. This creates an array that can hold 6 references to objects of type Shape.
Next, the code assigns different objects to each of the 6 array elements using the syntax `shapes[index] = new Object();`. In this case, we add a generic shape object to the first element, a circle object to the second, a square object to the third, another circle object to the fourth, another square object to the fifth, and a third circle object to the sixth.
By using this code, you have created an array of shape references and filled it with an assortment of circle and square objects.
To create an array of 6 shape references named `shapes` and fill it with an assortment of shape, circle, and square objects, you can follow these steps:
1. Define the base `Shape` class along with derived classes `Circle` and `Square`.
2. Create an array of `Shape` references named `shapes` with a size of 6.
3. Instantiate various `Circle` and `Square` objects and assign them to the elements in the `shapes` array.
Here's an example in Java:
```java
// Define the base Shape class
abstract class Shape {
   // Shape properties and methods
}
// Define the Circle class derived from Shape
class Circle extends Shape {
   // Circle properties and methods
}
// Define the Square class derived from Shape
class Square extends Shape {
   // Square properties and methods
}
public class Main {
   public static void main(String[] args) {
       // Create an array of 6 Shape references named shapes
       Shape[] shapes = new Shape[6];
       // Fill the array with an assortment of Circle and Square objects
       shapes[0] = new Circle();
       shapes[1] = new Square();
       shapes[2] = new Circle();
       shapes[3] = new Square();
       shapes[4] = new Circle();
       shapes[5] = new Square();
   }
}
```
This code creates an array of 6 shape references named `shapes` and fills it with an assortment of `Circle` and `Square` objects.

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The volume of the given solid over the indicated region of integration, f(x,y) = 2x + 2y - 5 and R = {(x,y): -4 ≤ x ≤ 1, 3 ≤ y ≤ 6}, is 63 units³.

To find the volume, we will integrate the function f(x,y) over the region R. Follow these steps:

1. Set up the double integral: ∬R (2x + 2y - 5) dA

2. Set up the limits of integration: ∫(from -4 to 1) ∫(from 3 to 6) (2x + 2y - 5) dy dx
3. Integrate with respect to y: ∫(from -4 to 1) [(2xy + 2y²/2 - 5y)] (evaluated from 3 to 6) dx
4. Simplify and evaluate: ∫(from -4 to 1) [6x + 18 - 5(6-3)] dx
5. Integrate with respect to x: [3x² + 18x - 15x] (evaluated from -4 to 1)
6. Simplify and evaluate: (3 + 18 - 15) - (-12 + 72 + 60)
7. Calculate the final result: 6 - (-30) = 36

The volume of the region is 63 units³.

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Greg receives $1100, Nigel receives $825, and Mike receives $1375 from the sale of the boat.

Greg lost $550 and Mike lost $825.

Nigel made the smallest loss, which was $275.

a) Let's first find the total amount paid by the three of them:

$1400 + $1050 + $1750 = $4200

Since they share the money from the sale of the boat in the same ratio as they paid for it, we need to find the total of the ratio:

$1400 + $1050 + $1750 = $4200

$3300/$4200 = $11/14

Now we can find how much each of them receives from the sale of the boat by multiplying the total amount by their ratio:

Greg: ($11/14) x $1400 = $1100

Nigel: ($11/14) x $1050 = $825

Mike: ($11/14) x $1750 = $1375

Therefore, Greg receives $1100, Nigel receives $825, and Mike receives $1375 from the sale of the boat.

b) To find out how much more money Mike lost from the sale of the boat than Greg, we need to calculate their individual losses.

We know that the boat was sold for $3300, and that the total cost was $4200. Therefore, their total loss is:

$4200 - $3300 = $900

To calculate their individual losses, we need to use the same ratio as before:

Greg: ($11/14) x $4200 - $1400 = $550

Mike: ($11/14) x $4200 - $1750 = $825

Therefore, Greg lost $550 and Mike lost $825.

To find out how much more money Mike lost than Greg, we subtract Greg's loss from Mike's loss:

$825 - $550 = $275

Therefore, Mike lost $275 more than Greg from the sale of the boat.

c) To find out who made the smallest loss from the sale of the boat, we need to calculate their individual losses again:

Greg: ($11/14) x $4200 - $1400 = $550

Nigel: ($11/14) x $4200 - $1050 = $275

Mike: ($11/14) x $4200 - $1750 = $825

Therefore, Nigel made the smallest loss, which was $275.

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Answer:

7. the solutions to the equation 2x^2 + x - 21 = 0 are x = -7/2 and x = 3

To factor the quadratic equation, we need to find two numbers that multiply to -42 and add up to 1. These numbers are 7 and -6. We can write:

2x^2 + x - 21 = 0

2x^2 + 7x - 6x - 21 = 0

(2x^2 + 7x) - (6x + 21) = 0

x(2x + 7) - 3(2x + 7) = 0

(2x + 7)(x - 3) = 0

Setting each factor equal to zero gives:

2x + 7 = 0 or x - 3 = 0

Solving for x, we get:

2x = -7,

x = -7/2 or x = 3

8. It took the receiver about 1.1 seconds to catch the ball.

Rearrange the equation to standard form, which is in the form of ax^2 + bx + c = 0:

-4.9t^2 + 7.5t - 0.3 = 0

Now we can use the quadratic formula to solve for t:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

Substituting in the values, we get:

t = (-7.5 ± sqrt(7.5^2 - 4(-4.9)(-0.3))) / 2(-4.9)

t = (-7.5 ± sqrt(56.25 - 5.88)) / (-9.8)

t = (-7.5 ± sqrt(50.37)) / (-9.8)

t ≈ 1.1 seconds or t ≈ 0.06 seconds

Since the time can't be negative, we can discard the negative solution.

The series ₙ=₁Σ∞ (n²+n+6) / (n⁴ + n²) is convergent.

Comparison test:

Let Σuₙ and Σvₙ be the two series of positive real numbers and there is a natural number in such that uₙ ≤ vₙ for all n ≥ m, k being a fixed positive number. Then,

i) Σuₙ is convergent if Σvₙ is convergent.

ii) Σvₙ is divergent if Σuₙ is divergent.

Given, ₙ=₁Σ∞ ( 6ⁿ⁺¹) / (5ⁿ -6 )

Let  ₙ=₁Σ∞ vₙ  be the given series where vₙ = ( 6ⁿ⁺¹) / (5ⁿ -6 )

Now, we have  ( 6ⁿ⁺¹) / (5ⁿ -6 ) > (6/5)ⁿ , ∀ n ≥ 2.

uₙ < vₙ  , where uₙ = (6/5)ⁿ.

By geometric series Σ(6/5)ⁿ is divergent.

Therefor, the series diverges by the comparison test. Each term is greater than that of a divergent geometric series.

Comparison test(limit form)

Let Σuₙ and Σvₙ be two series of positive real numbers and [tex]\lim_{n \to \infty} \frac{u}{v} = l\\[/tex], where l is a non-zero finite number. Then the two series  Σuₙ and Σvₙ converge and diverge together.

Given, ₙ₋₁Σ∞ (n²+n+6) / (n⁴ + n²)

Let ₙ₋₁Σ∞ uₙ be the given series

Then, uₙ = (n²+n+6) / (n⁴ + n²)

Let, vₙ = 1/(n²)

Then, [tex]\lim_{n \to \infty} \frac{u}{v\\}[/tex] = (n²+n+6). n² / (n⁴ + n²)

= [tex]\lim_{n \to \infty} \\[/tex] [n⁴ (1 + 1/n + 6/n²)] / n⁴ (1 + 1/n²)

= 1 (non-zero finite number)

Since, Σ (1/n²) convergent by p-series[p = 2>1]

Therefore, By comparison test

Σuₙ is convergent.

Hence, the series ₙ=₁Σ∞ (n²+n+6) / (n⁴ + n²) is convergent.

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Complete question:

Determine whether the series converges or diverges.

ₙ=₁Σ^∞ ( 6ⁿ⁺¹) / (5ⁿ -6 )

The region in the plane for points with polar coordinates 2 < r < 3 and 5π/3 ≤ θ ≤ 7π/3.


Identify the range of r and θ: In this case, 2 < r < 3 and 5π/3 ≤ θ ≤ 7π/3. Draw the polar coordinate plane (with an origin, labeled "O") and radial lines representing the angles θ.Mark the angle 5π/3 on the plane, which is located in the fourth quadrant (5π/3 = 300°). Draw a radial line from the origin to represent this angle.

Mark the angle 7π/3 on the plane, which is also located in the fourth quadrant (7π/3 = 420°, but since 360° brings us back to the origin, it is equivalent to 60°). Draw a radial line from the origin to represent this angle.Draw two concentric circles around the origin with radii 2 and 3. These represent the range of r values.

Shade the region in the plane that is bounded by the radial lines, as well as the circles with radii 2 and 3. This is the region consisting of points whose polar coordinates satisfy the given conditions.

Your sketch should now show the region in the plane for points with polar coordinates 2 < r < 3 and 5π/3 ≤ θ ≤ 7π/3.

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Answer:

(0;7)

Step-by-step explanation:

The graph will cross the y-axis when x is zero

Replace x with 0 in the given function:

[tex]7 \times {0}^{2} + 2 \times 0 + 7[/tex]

[tex]0 + 0 + 7 = 7[/tex]

x = 0,

y = 7

To find the associated z-score for each standard normal area, you can use the Excel function =NORM.S.INV().

a. For the highest 10 percent, use the formula =NORM.S.INV(1-0.10), which gives a z-score of 1.2816.
b. For the lowest 50 percent, use the formula =NORM.S.INV(0.50), which gives a z-score of 0.0000.
c. For the highest 7 percent, use the formula =NORM.S.INV(1-0.07), which gives a z-score of 1.4758.


The Excel function =NORM.S.INV() calculates the inverse of the standard normal cumulative distribution function for a given probability. The function takes one argument, which is the probability of the area to the left of the desired z-score.

To find the z-score for the highest X percent, use the formula =NORM.S.INV(1-X). For the lowest X percent, simply input the probability X as the argument in the function. Finally, round the resulting z-scores to four decimal places.

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Answer:

To find a basis for W, we need to find a set of linearly independent matrices in W that span W.

Let's first rewrite the condition for A to be in W:

A = [a a - c

b 2a + c]

We can rewrite A in terms of a linear combination of matrices as follows:

A = a [1 1

0 2] + b [0 -1

1 0] + c [0 1

0 1]

Therefore, any matrix A in W can be written as a linear combination of the three matrices:

B1 = [1 1

0 2],

B2 = [0 -1

1 0],

B3 = [0 1

0 1]

We just need to check that these three matrices are linearly independent. To do this, we set up the equation

c1 B1 + c2 B2 + c3 B3 = 0

where c1, c2, c3 are scalars. This gives the system of linear equations

c1 = 0

c2 - c3 = 0

c1 + c2 + c3 = 0

The solution to this system is c1 = 0, c2 = c3, and any value for c2. This means that the three matrices are linearly independent, and hence they form a basis for W.

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Step-by-step explanation:

The matrix A of the linear transformation is:

A = | -1/2 √3/2|

| -√3/2 -1/2 |

To find the matrix A of the linear transformation T that rotates any vector through an angle of 120 degrees in the clockwise direction, we can use the standard matrix formula for rotating a vector in two dimensions:

|cos θ - sin θ|

|sin θ cos θ|

where θ is the angle of rotation in radians. Since we want to rotate through an angle of 120 degrees clockwise, which is equivalent to -2π/3 radians, we can plug in θ = -2π/3 to get:

|cos(-2π/3) - sin(-2π/3)|

|sin(-2π/3) cos(-2π/3)|

We can simplify this using the trigonometric identities cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), to get:

| -1/2 √3/2|

| -√3/2 -1/2 |

Therefore, the matrix A of the linear transformation T that rotates any vector through an angle of 120 degrees in the clockwise direction is:

A = | -1/2 √3/2|

| -√3/2 -1/2 |

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The solution is : The value is, f(x)·g(x) = x³ +x² -2x -2

Given that,

Let f(x)=x+1

and g(x)= x^2 – 2.

Substitute the function definitions and simplify.

 f(x)·g(x)

 = (x +1)(x² -2)

 = x(x² -2) +1(x² -2)

 = x³ -2x +x² -2

so, we get,

 f(x)·g(x) = x³ +x² -2x -2

The value is, f(x)·g(x) = x³ +x² -2x -2

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complete question:

Let f(x)=x+1 and g(x)= x^2 – 2. Find f (x)· g(x)

To find the p-value for the hypothesis test, we first need to state the null and alternative hypotheses:

Null Hypothesis: The mean number of alcoholic drinks that students at this liberal arts college have in a typical week is equal to the mean number of alcoholic drinks that U.S. college students in general have, which is 4.73.

Alternative Hypothesis: The mean number of alcoholic drinks that students at this liberal arts college have in a typical week is different from the mean number of alcoholic drinks that U.S. college students in general have, which is 4.73.

We can then use a t-test to determine if the difference between the sample mean (4.21) and the hypothesized population mean (4.73) is statistically significant. The formula for the t-test is:

t = (x - μ) / (s / sqrt(n))

where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the values from the problem, we get:

t = (4.21 - 4.73) / (3.43 / sqrt(50))
t = -1.58

Using a t-table with degrees of freedom (df) = 49 (n - 1), we find that the two-tailed p-value for t = -1.58 is 0.1192.

Therefore, the p-value for the hypothesis test is 0.1192, rounded to 4-decimal places.
In order to find the p-value for this hypothesis test, we first need to state the null hypothesis (H₀) and alternative hypothesis (H₁).

H₀: μ = 4.73 (The mean number of drinks at the college is equal to the national mean)
H₁: μ ≠ 4.73 (The mean number of drinks at the college is different from the national mean)

Now, let's use the provided sample data to perform a two-tailed t-test. We have:

Sample size (n) = 50
Sample mean (x) = 4.21
Population mean (μ) = 4.73
Sample standard deviation (s) = 3.43

Calculate the t-statistic:
t = (x - μ) / (s / √n)
t = (4.21 - 4.73) / (3.43 / √50)
t = -0.52 / 0.485
t ≈ -1.072

With n-1 = 49 degrees of freedom and a t-statistic of -1.072, we will find the p-value using a t-distribution table or calculator. The p-value for a two-tailed test will be:

p-value = 2 * P(T < -1.072)
p-value ≈ 0.2888

So, the p-value for this hypothesis test is approximately 0.2888 when rounded to 4 decimal places.

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The domain of f(x,y) = √x + √y is the set of all points (x,y) in the first quadrant of the xy-plane, where both x and y are greater than or equal to zero. So, correct option is E.

The domain of a function is the set of all possible values of the input variables for which the function is defined. In the case of the function f(x,y) = √x + √y, the values of x and y must be such that the expression under the square root is non-negative. This means that x and y must be greater than or equal to zero.

Therefore, the domain of f(x,y) is the set of all points (x,y) in the first quadrant, where both x and y are greater than or equal to zero. The first quadrant is the region of the xy-plane where x and y are both positive.

Therefore, the correct answer is (e) the first quadrant.

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Complete question is:

The domain of the function f(x,y) = √x + √y  is

a. all of the xy-plane

b. the area inside a parabola

c. the union of two intervals

d. the first and third quadrants

e. the first quadrant

The expression that describes the relationship is as follows:

D. The number of sheets of paper is 50 times the number of packages

The number of sheet of paper in x packages is described by the rule y = 50x.

Therefore, the verbal description that describes the relationship can be represented as follows:

y = 50x

where

Therefore, the number of sheets of paper is 50 times the number of packages describes the relationship.

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The expression x² - 6 + 16x + 30, when rewritten in standard form is x² - 6 + 16x + 30 and factored as (x + 4)(x + 12)

What is an algebraic expression?

An algebraic expression is a mathematical phrase that can contain numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

To rewrite x² - 6 + 16x + 30 in standard form, we need to combine the terms that involve x^2 and x, respectively, and also simplify the constant terms.

x² - 6 + 16x + 30

= x² + 16x + 24 (combining like terms and simplifying constants)

To factor x² + 16x + 24, we need to find two numbers whose product is 24 and whose sum is 16, the coefficient of the x-term. These numbers are 4 and 12, since 4 * 12 = 48 and 4 + 12 = 16.

Therefore, we can write:

x² + 16x + 24 = (x + 4)(x + 12)

So the expression x² - 6 + 16x + 30, when rewritten in standard form and factored, is:

x² - 6 + 16x + 30 = (x + 4)(x + 12)

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option D (4, 7, 10, 13, ...) is the number pattern that uses the rule "add 3."

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.

A number pattern is a sequence of numbers that follows a certain rule or pattern. In this case, we are given four number patterns (A, B, C, and D) and asked to identify which one uses the rule "add 3."

Let's take a closer look at each of the patterns:

A: 2, 6, 18, 48, ...

To determine the rule for this sequence, we need to look at the relationship between each pair of consecutive numbers. We can see that each number in the sequence is obtained by multiplying the previous number by 3.

B: 3, 7, 11, 15,...

Similarly, to determine the rule for this sequence, we need to look at the relationship between each pair of consecutive numbers. We can see that each number in the sequence is obtained by adding 4 to the previous number.

C: 3, 9, 27, 54,...

To determine the rule for this sequence, we need to look at the relationship between each pair of consecutive numbers. We can see that each number in the sequence is obtained by multiplying the previous number by 3. However, this pattern starts with 3 instead of 2, so it is a variation of pattern A.

D: 4, 7, 10, 13,...

Finally, we come to pattern D. We can see that each number in the sequence is obtained by adding 3 to the previous number.

Therefore, option D (4, 7, 10, 13, ...) is the number pattern that uses the rule "add 3."

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The quotient of the division operation (12x³ - 11x² + 9x + 18) / (4x + 3) is 3x² - 5x + 6

From the question we are to determine the quotient in the given division operation

The given division operation is:

(12x³ - 11x² + 9x + 18) / (4x + 3)

The quotient of the given division operation can be determine as follows

                     3x² - 5x + 6

4x + 3 | 12x³ - 11x² + 9x + 18

           12x³ + 9x²  

                    -20x² + 9x

                    -20x² - 15x  

                                 24x + 18

                                 24x + 18

                                            0

Hence, the quotient of the division operation is 3x² - 5x + 6

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Step-by-step explanation:

2/3 = 66.666666666666.......

1/6= 16.66666666667

yeap, the midpoint is the center of it and half the distance between those two points is the radius.

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{-2}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{7}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 4 -2}{2}~~~ ,~~~ \cfrac{ 7 +5}{2} \right) \implies \left(\cfrac{ 2 }{2}~~~ ,~~~ \cfrac{ 12 }{2} \right)\implies \stackrel{ center }{(1~~,~~6)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-2}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{7})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{ diameter }{d}=\sqrt{(~~4 - (-2)~~)^2 + (~~7 - 5~~)^2} \implies d=\sqrt{(4 +2)^2 + (7 -5)^2} \\\\\\ d=\sqrt{( 6 )^2 + ( 2 )^2} \implies d=\sqrt{ 36 + 4 } \implies d=\sqrt{ 40 }~\hfill \stackrel{radius}{\cfrac{\sqrt{40}}{2}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{1}{h}~~,~~\underset{6}{k})}\qquad \stackrel{radius}{\underset{\frac{\sqrt{40}}{2}}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - 1 ~~ )^2 ~~ + ~~ ( ~~ y-6 ~~ )^2~~ = ~~\left( \frac{\sqrt{40}}{2} \right)^2\implies \boxed{(x-1)^2+(y-6)^2=10}[/tex]

Answer:32.9

Step-by-step explanation:

Hope this helps! :)

Answer:

Diameter = 103.35 cm

Step-by-step explanation:

Pizza = circle

Area of a Circle = π * r²

π = 3.1416

r = radius

Area of a pizza = 850cm²

850 = 3.1416 * r²

850/3.1416 = r²

270.56 = r²

√270.56 = √r²

16.45 = r

radius = 16.45  (aprox)

diameter of a pizza = 2π r

diameter of a pizza = 2*3.1416*16.45

diameter of a pizza = 103.35 cm

Answer:

Step-by-step explanation:

If 1.5 inches on the map represent a distance of 8.1 miles in reality, we can use the proportion:

1.5 inches / 1 inch = 8.1 miles / x miles

where x is the actual distance represented by 1 inch on the map, and is the scale of the map we are looking for.

We can solve for x by cross-multiplying:

1.5x = 8.1

x = 8.1 / 1.5

x = 5.4

Therefore, the scale of the map is 1 inch represents 5.4 miles.

Answer: Number selected;  26

% score = 86.67%

Grade is a B

Step-by-step explanation:

Assuming a person selects a number from 21-29

Number seleted; 26

Say that person A gets 26/30 on his test

SN= selected number , 26

SN/Total score x 100%

26/30 x 100%

= 0.8667 x 100%

% score is 86.67%

The dimensions that minimize the amount of cardboard used are,

⇒ Length = 80 cm, Width = 40 cm, Height = 20 cm.

label the dimensions of the rectangular cardboard box as length (L), width (W), and height (H).

We can use the formula for volume of a rectangular prism to get,

⇒ V = LWH

       = 32,000 cm³

Now, we want to minimize the amount of cardboard used to make this box.

Since there is no lid, we can assume that the cardboard is cut from a single piece and then folded into a box shape.

In this case,

the amount of cardboard used will be proportional to the surface area of the cardboard sheet. So, our goal is to minimize the surface area (SA) of the cardboard box.

We can find the surface area of the box using the formula,

⇒ SA = 2LW + 2LH + 2WH

Now, we need to express SA in terms of a single variable so we can differentiate it and find the minimum value.

To do this, we can use the volume equation to solve for one of the variables. For example, we can solve for L,

⇒L = 32000/WH

Substituting this into the equation for SA, we get,

⇒ SA = 2W(32000/WH) + 2H(32000/WH) + 2WH SA

          = 64000/W + 64000/H + 2WH

Now, we can differentiate SA with respect to either W or H and set the result equal to zero to find the minimum value.

Differentiate with respect to W:

⇒ dSA/dW = -64000/W² + 2H

Setting this equal to zero and solving for W, we get:

⇒ W = √(32000/H)

Substituting this back into the volume equation, we can solve for L:

⇒ L = sqrt(32000H)

Now, we can substitute these expressions for L and W into the equation for SA and simplify,

⇒ SA = 64000/sqrt(H) + 64000/sqrt(L) + 2HL SA

         = 64000/sqrt(H) + 64000/sqrt(32000/H) + 2Hsqrt(32000/H)

We can simplify this further by using a substitution u = √(H):

⇒ SA(u) = 64000/u + 3200u + 128000/u

Now, we can differentiate SA(u) with respect to u and set it equal to zero to find the minimum value,

⇒ dSA/du = -64000/u² + 3200 - 128000/u² = 0

⇒ 128000/u² = (64000/u^2) - 3200 u² = 20

Substituting back in for H and solving for L and W, we get:

⇒ H = 20 cm² L

       = √(32000H)

       = 80 cm W

       = √(32000/H)

        = 40 cm

Therefore, the dimensions that minimize the amount of cardboard used are,

⇒ Length = 80 cm, Width = 40 cm, Height = 20 cm.

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Answer:

Question 7: The correct graph to represent the data is a bar graph with the title favorite sport and the x-axis labeled sport and the y-axis labeled number of students, with the first bar labeled basketball going to a value of 17, the second bar labeled baseball going to a value of 12, the third bar labeled soccer going to a value of 27, and the fourth bar labeled tennis going to a value of 44.

Question 8: The correct circle graph to represent the data is a circle graph titled New York City visitor's transportation, with five sections labeled walk 48 percent, bus 8 percent, car service 18 percent, bicycle 4 percent, and subway 22 percent.

Question 9: The best prediction about the scoops of ice cream the college will need is that the college will have about 640 students who prefer ice cream. This prediction is based on finding the ratio of students who prefer ice cream in the sample and using that ratio to estimate the number of students who prefer ice cream in the whole population.

Question 10: Burger Quick typically has more wait time because it has a larger interquartile range (IQR), which represents the middle 50% of the data. Even though Burger Quick's median is larger, it's not necessarily a better measure of central tendency than Super Fast Food's median. Nonetheless, since both are skewed to the right, using their means, which are susceptible to being influenced by outliers, could be misleading.

Question 11: The most appropriate graphical representation for the data is a histogram because it shows the frequency distribution and range of the data. A circle chart may not be appropriate since there are more than three categories of data, while a line plot does not work well with this type of data. Stem-and-leaf plots can be used if there is only a small number of data points, but a histogram is better when the number of data points is relatively large.

The chi-square distributions are skewed in general, and they are skewed to the right.

The maximum number of logically independent values, or values with the ability to fluctuate, in the data sample is referred to as degrees of freedom.

Degrees of freedom are frequently used in statistics to describe many types of hypothesis testing, such as the chi-square test.

Degrees of freedom are important for finding critical cutoff values for inferential statistical tests. Depending on the type of the analysis you run, degrees of freedom typically relate the size of the sample.

Thus, chi-square distributions is right-skewed.

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when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one.  Check the picture below.

[tex]\cfrac{y}{5}=\cfrac{11}{y}\implies y^2=55\implies y=\sqrt{55}\implies y\approx 7.4 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{x}{5}=\cfrac{16}{x}\implies x^2=80\implies x=\sqrt{80}\implies x\approx 8.9[/tex]

Answer:  Ordinary Interest Method (360 Days)

Calculating Simple Interest using 360 days per year in time.

Time = Exact Number of Days

360

Since banks commonly use the ordinary interest method, it is known as the bankers rule.

Step-by-step explanation: