Let a sphere of radius r be cut by a plane, thereby forming a segment of height h. Show that the volume of this segment is (∏h2(3r-h))/3

A. To find the correlation coefficient, we can use the formula:

r = (n * Σxy - Σx * Σy) / √ ((n * Σx² - (Σx)²) * (n * Σy² - (Σy)²))

Where n is the number of data points.

Plugging in the values for the given data set:

r = (12 * 712.94 - 1451.9 * 511.6) / √ ((12 * 1297.14 - 1451.9²) * (12 * 636.35 - 511.6²))

r = -0.636

So, the correlation coefficient is -0.636.

B. To find the proportion of the variation in y that can be explained by the variation in x, we can use the formula:

r² = r²

r² = (-0.636) ²

r² = 0.40496

So, 40.5% of the variation in y can be explained by the variation in the values of x.

C. To find the regression line,we can use the formula:

y = a + bx

Where a is the y-intercept and b is the slope of the line.

To find b, we can use the formula:

b = (n * Σxy - Σx * Σy) / (n * Σx² - (Σx)²)

Plugging in the values for the given data set:

b = (12 * 712.94 - 1451.9 * 511.6) / (12 * 1297.14 - 1451.9²)

b = -0.856

To find a, we can use the formula:

a = (Σy - b * Σx) / n

Plugging in the values for the given data set:

a = (511.6 - -0.856 * 1451.9) / 12

a = 55.54

So, the regression line is:

y = 55.54 - 0.856x

The negative slope of -0.856 means that as x increases, y decreases, which is consistent with the negative correlation coefficient of -0.636. The y-intercept of 55.54 means that when x is 0, y is predicted to be 55.54.

Complete question:

Run a regression analysis on the following bivariate set of data with y as the response variable.

x y

43.8 54.1

41.3 51.2

35.3 60.1

48.1 44.9

42.8 51.8

44.7 50.8

39.4 56.6

38.2 56.6

40.7 53.5

45.3 51.1

45 46.7

41.1 52.2

A. Find the correlation coefficient and report it accurate to three decimal places.

r = ________________

B. What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place.  (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)

r² =_______________ %

C. Based on the data, calculate the regression line (each value to three decimal places)

y =___________ x + _______________

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Charisse needs to stain an area of 174 square feet.

The total surface area of the shed's walls that needs to be stained is the sum of the areas of the four walls, minus the area of the front door.

The area of one wall is 8 feet × 6 feet = 48 square feet.

Since there are 4 walls, the total area of all walls is 48 square feet × 4 = 192 square feet.

The area of the front door is 3 feet  × 6 feet = 18 square feet

Now we can subtract the area of the door from the total area of all walls:

192 square feet - 18 square feet = 174 square feet

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There can be one valid triangle created from the given conditions of side a being 5 and angles b and c being 30 and 50 degrees respectively.

To calculate how many triangles can be created from two angles and a side, the user must click the “calculator” tab. Once this is done, it is important to click the “clear” button to begin a new calculation. After this, the user must enter the side (a) under the “sides” section, and the angles (b and c) under the “angles” section. This example used 5 for the side, 30 for angle b, and 50 for angle c. Once the information is entered, the user must click “calculate”. This will result in a valid triangle being created from the given conditions. It is important to note that if the user entered different values, a different result may be obtained. Therefore it is important to ensure that the values entered are the desired ones. Additionally, if the sum of the angles entered are not equal to 180 degrees, then no valid triangle can be created.

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Answer:

x = 121°

Step-by-step explanation:

the shape is a heptagon (7 sides). the interior angles in a heptagon will always add up to 900°, so all you need to do is add all the known angles and subtract from 900.

140 + 133 + 145 + 117 + 119 + 125 = 779

900 - 779 = 121

this means that x = 121°

hope this helped, good luck!

To simplify the inequality, we need to perform the same mathematical operation on both sides of the inequality.

V-7 > 24

Add 7 to both sides of the inequality:

V > 31

So, the simplified inequality is V > 31.

The length of the pile of the piece of woods put together by Luke would be = 5.8ft

A pile of objects such as wood is defined as the combination of two or more different parts of the object in an orderly form.

The number of woods stacked by Like = 7

The length of each wood = 10/12ft

Therefore the length of the whole pile of wood ;

= 7×10/12

= 70/12

= 5.8ft

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Algorithm 2 is more efficient than Algorithm 1 because it only needs to loop through the numbers up to upper/11 instead of upper. This means that it will take fewer iterations to reach the result, making it more efficient than Algorithm 1.

Algorithm 1 will make upper/11 more checks than Algorithm 2, resulting in a slower runtime. Additionally, Algorithm 2 avoids the need to check the modulus of each number, further reducing the computation time. Algorithm 2 only needs to loop through the multiples of 11, which are always 11 apart from each other, so it can skip over the other numbers between them.

This makes Algorithm 2 much faster than Algorithm 1 because it is only looping through a fraction of the numbers.

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The surface area of the hexagonal pyramid is S = 575.9 inches²

The total surface area is the summation of the areas of the base and the three other sides. A = B + ( 1/2 ) ( P x h ), where B is the area of the base of the pyramid, P is the perimeter of the base, and h is the slant height of the pyramid

Surface Area of Pyramid = B + ( 1/2 ) ( P x h )

Given data ,

Let the surface area of the hexagonal pyramid be S

Now , the equation will be

The base of the hexagonal pyramid a = 10 inches

The height of the lateral face h = 6 inches

The base area of the pyramid = ( 3√3 / 2 )a²

The lateral surface area of the pyramid = 3a √ ( ( 3a²/4 ) + h² )

So , The total surface area of hexagonal pyramid is S

S = ( 3√3 / 2 )a² + 3a √ ( ( 3a²/4 ) + h² )

Substituting the values in the equation , we get

The base area of the pyramid = 259.8 inches²

The lateral surface area of the pyramid = 3 ( 10 ) √ ( 3 ( 100/4 ) + 36 )

On simplifying the equation , we get

The lateral surface area of the pyramid = 316.1 inches²

Total surface area of hexagonal pyramid S = 259.8 inches² + 316.1 inches²

The total surface area of hexagonal pyramid S = 575.9 inches²

Hence , the surface area of the pyramid is 575.9 inches²

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The most common level of confidence is 95%, however 90% or 99% have also been used on occasion.

A series of observations that are not reliant on any other samples or data constitute an independent random sample. Random samples selected without regard to sequencing. So, choice C is the proper one.

The essential points that are highlighted below are the followings:

Because the difference between the sample proportions G minus H greater population, according to the question. Due to the interval's all-positive values, it seems expected that district G will have more supporters than district H.

Assurance interval A confidence interval (CI) is a range of estimates for an unknown parameter that is determined by a lower bound and an upper bound. At a certain level of confidence, the interval is calculated.

The most typical level of confidence is 95%, however higher levels, such 90% or 99%, are occasionally employed.

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Congruency is said to be defined on two triangles when they have exact same length of sides and equal angles between them. The shape and size are supposed to be the same even after rotation or flipping of triangles for them to remain congruent.

There are 4 criteria to test the congruency of the triangle and they are as follows:

In this set of triangles, (ABC, def) AB=DE, AC=DF, and ∠ABC= ∠DEF i.e., two sides and one angle are equal but it is not congruent because according to SAS criteria, the angle should be between AB and AC (or DE and DF)

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Answer: X[tex]\geq[/tex]-4

Step-by-step explanation:

4x-5[tex]\leq[/tex] 6x + 3

-2x [tex]\leq[/tex] 3 + 5

-2x[tex]\leq[/tex] 8

X = [tex]\geq[/tex] - 8/2

The solution set for the inequality -4≤-2(y-1)<2 is the set of all y-values that make the inequality true.

To graph the solution set, we can begin by plotting the inequality as an inequality on the y-axis, and then identifying the solutions that make the inequality true:

First, we can simplify the left side of the inequality: -4≤-2(y-1)

Next, we can solve for y by isolating y: -4/2≤y-1

Then we can add 1 to both sides of the inequality: -2≤y

Now, we can graph the inequality y ≥ -2, which is a line that is equal to or greater than -2.

On the right side, we have -2(y-1)<2

2(y-1)>-2

y-1>-1

y>-1

So we can graph the inequality y>-1, which is a line that is greater than -1.

So the solution set is the region above the line y=-2 and below the line y=-1. The region is a strip between two lines.

It is important to note that the solution set doesn't include the values of the lines, as the inequality is strict, not inclusive.

The value of y² is 1/5 ln(3+x¹⁰) + C

What is the separation of variables?

Separation of variables, in mathematics, is any of several strategies for rewriting equations so that each of two variables appears on a different side of the problem. These strategies are used to solve ordinary and partial differential equations.

Here, we have

Given: (3+^10)(/) = ^9/

We have to solve the differential equation by using the separation of variables.

Rewrite as y dy = x⁹/3+x¹⁰ dx

As (3+x¹⁰)' = 10x⁹, we solve

y²/2 = 1/10 ln(3+x¹⁰) + C

y² = 1/5 ln(3+x¹⁰) + 2C

Since C is any constant, y²  = 1/5 ln(3+x¹⁰) + C

Hence, the value of y² is 1/5 ln(3+x¹⁰) + C

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Answer:5(3+1); 20

Step-by-step explanation:

5x3 is 15 and 5x1 is 5 which is equivalent to 15+5, then add to get 20.

The answers contain a. 5  b. [tex]y=-\frac{3}{4} x+\frac{1}{2}[/tex]  c. 148.5 d. 1/7

a. f"(2)

f"(x) = df'(x)/dx = d(sin(πx) + x² +3)/dx = cos(πx) + 2x

f"(2)=cos(π × 2) + 2 × 2

f"(2)=cos(2π) + 4

f"(2)=1 + 4

f"(2)=5

b. Equation for the line tangent to the graph of y = 1/f(x) at x = 0

We first find f(x) by integrating f'(x)

f(x) = ∫f'(x)dx = ∫(sin(πx) + x² +3)dx = -cos(πx)/π + x³/3 +3x + C

f(0) = 2 so,

2 = -cos(π × 0)/π + 0³/3 +3 × 0 + C

2 = -cos(0)/π + 0 + 0 + C

2 = -1/π + C

C = 2 + 1/π

f(x) = -cos(πx)/π + x³/3 +3x + 2 + 1/π

f(x) = [1-cos(πx)]/π + x³/3 +3x + 2

y = 1/f(x) = 1/([1-cos(πx)]/π + x³/3 +3x + 2)

The tangent to y is thus dy/dx

dy/dx = d1/([1-cos(πx)]/π + x³/3 +3x + 2)/dx

dy/dx = -([1-cos(πx)]/π + x³/3 +3x + 2)⁻²(sin(πx) + x² +3)

at x = 0,

dy/dx = -([1-cos(π × 0)]/π + 0³/3 +3 × 0 + 2)⁻²(sin(π × 0) + 0² +3)

dy/dx = -([1-cos(0)]/π + 0 + 0 + 2)⁻²(sin(0) + 0 +3)

dy/dx = -([1 - 1]/π + 0 + 0 + 2)⁻²(0 + 0 +3)

dy/dx = -(0/π + 2)⁻²(3)

dy/dx = -(0 + 2)⁻²(3)

dy/dx = -(2)⁻²(3)

dy/dx = -3/4

At x= 0,

y = 1/([1-cos(π × 0)]/π + 0³/3 +3 × 0 + 2)

y = 1/([1-cos(0)]/π + 0 + 0 + 2)

y = 1/([1 - 1]/π + 2)

y = 1/(0/π + 2)

y = 1/(0 + 2)

y = 1/2

So, the equation of the tangent at (0, 1/2) is

[tex]\frac{y-\frac{1}{2} }{x-0} =-\frac{3}{4} \\\\y-\frac{1}{2} =-\frac{3}{4} x\\\\y=-\frac{3}{4} x+\frac{1}{2}[/tex]

c. If g(x) = f (√(3x² + 4). Find g'(2)

g(x) = f (√(3x² + 4) = [1-cos(π√(3x² + 4)]/π + √(3x² + 4)³/3 +3√(3x² + 4) + 2

g'(x) = [3xsinπ√(3x² + 4) + 18x(3x² + 4) + 9x]/√(3x² + 4)

g'(2) = [3(2)sinπ√(3(2)² + 4) + 18(2)(3(2)² + 4) + 9(2)]/√(3(2)² + 4)

g'(2) = [6sinπ√(12 + 4) + 36(12 + 4) + 18]/√12 + 4)

g'(2) = [6sinπ√(16) + 36(16) + 18]/√16)

g'(2) = [6sin4π + 576 + 18]/4)

g'(2) = [6 × 0 + 576 + 18]/4)

g'(2) = [0 + 576 + 18]/4)

g'(2) = 594/4

g'(2) = 148.5

d. If h is the inverse function of f. Find h' (2)

If h(x) = f⁻¹(x)

then h'(x) = 1/f'(x)

h'(x) = 1/(sin(πx) + x² +3)

h'(2) = 1/(sin(π2) + 2² +3)

h'(2) = 1/(sin(2π) + 4 +3)

h'(2) = 1/(0 + 4 +3)

h'(2) = 1/7

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There are 15 possible sums less than 15 that can be made by taking out one to three balls from the chest.

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The Minimum for quadratic function f(x) is smaller than g(x)

and it is -5.

What is a quadratic function?

Quadratic function is also a second-degree polynomial function. The graph of a quadratic function is a parabola. The parabola opens upwards if a graph is made for the quadratic formula.

The point at which the function attains maximum or minimum value is the vertex of the quadratic function. When we say second degree, then the variable is raised to the second power like x^2.

A quadratic function is of the form f(x) = ax^2 + bx + c, where a, b, and c are the numbers with a not equal to zero.

Now,

As the we find minimum value by calculating the value for f(x) or g(x)

and from given values f(x) have minimum value of -5 and g(x) have minimum value of 3.

hence,

          Minimum for quadratic function f(x) is smaller than g(x)

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The side length of the original cube is 8.90 inches

Let x be the side length of the original cube. After cutting off the top, the remaining rectangular box has dimensions of x-5, x-5, and x.

The volume of this rectangular box is:

(x-5)(x-5)(x) = x³ - 10x² + 25x

x³ - 10x² + 25x = 135

x³ - 10x² + 25x -135 = 0

Use cubic formula.

x =8.90

Therefore, the side length of the original cube is 8.90 inches

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Answer:

Step-by-step explanation:

If its going by 4 each time then it should be 21 but for n its 6

There are 4 quarts in a gallon, and a gallon weighs 8.34 pounds, so a quart weighs 2.085 pounds.

The gallon is a volumetric unit that is used in both imperial and American customary measurements. In use right now are three different variations:

the imperial gallon (imp gal), defined as 4.54609 litres, which is or was used in the United Kingdom, Ireland, Canada, Australia, New Zealand, and some Caribbean countries; the US gallon (US gal), defined as 3.785411784 L,[1] (231 cubic inches), which is used in the US and some Latin American and Caribbean countries; and the US dry gallon ("usdrygal"), defined as 18 US bushel (exactly 4.40488377086 L).

gallon equals 4 quarts

2.1*4 =8.4

gallon of water weigh is 8.4

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The value of the composite function (f + g)(x) = 14x - 3

From the question, we have the following parameters that can be used in our computation:

f(x) = 2x - 1

g(x) = 12x - 2

The composite function (f + g)(x) is calculated as

(f + g)(x) = f(x) + g(x)

Substitute the known values in the above equation, so, we have the following representation

(f + g)(x) = 2x - 1 + 12x - 2

Evaluate the like terms

(f + g)(x) = 14x - 3

Hence, the composite function is (f + g)(x) = 14x - 3

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Complete question

Consider the functions f(x)=2x−1 g(x)=12x −2

Calculate (f + g)(x)

The rule for converting a binary number to  Decimal point is to multiply each binary digit by two to the power of its position from the right, and add all the resulting values together. For example, 42 in binary is 101010, so the conversion is 1*2^5 + 0*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 0*2^0 = 32 + 8 + 2 = 42.

1. Start with the rightmost binary digit (the ones place) and multiply it by 2^0.

2. Move to the next digit to the left (the twos place) and multiply it by 2^1.

3. Continue this process until you reach the leftmost binary digit.

4. Add all the products together to get the decimal equivalent.

For example, 42 in binary is 101010, so the calculations are: 0*2^0 + 1*2^1 + 0*2^2 + 1*2^3 + 0*2^4 + 1*2^5 = 0 + 2 + 0 + 8 + 0 + 32 = 42.

The rule for converting a binary number to decimal is to multiply each binary digit by two to the power of its position from the right, and add all the resulting values together. To do this, start with the rightmost binary digit (the ones place) and multiply it by 2^0. Then move to the next digit to the left (the twos place) and multiply it by 2^1. Continue this process until you reach the leftmost binary digit. Add all the products together to get the decimal equivalent. For example, 42 in binary is 101010, so the calculations are:

0*2^0 + 1*2^1 + 0*2^2 + 1*2^3 + 0*2^4 + 1*2^5 = 0 + 2 + 0 + 8 + 0 + 32 = 42.

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The volume of the squared pyramid is 9 cubic yards, so the correct option is D.

For a square pyramid of sidelength S (S is the sidelength of the square base), the volume is given by the formula:

V = (1/3)*S^3

Here we want to find the volume if S = 3 yards, so we just need to replace that in the formula above to get the volume of the sculpture, we will get:

V = (1/3)*(3 yd)^3

V = (1/3)*(27 yd^3)

V = 9 yd^3

The volume is 9 cubic yards, then the correct option is D.

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The range of the function h(x) is -

"all real numbers greater than or equal to -2​"

        [tex]${\displaystyle \forall x\in X,\forall y\in Y,\forall z\in Y,\quad ((x,y)\in R\land (x,z)\in R)\implies y=z.}[/tex]

        [tex]${\displaystyle \forall x\in X,\exists y\in Y,\quad (x,y)\in R.}[/tex]

Given is a table representing a function.

Range is defined as the set of all possible values of f(x). Therefore, all real numbers greater than or equal to -2​, will belong to the range of the function h(x).

Therefore, the range of the function h(x) is -

"all real numbers greater than or equal to -2​"

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The value of X that is less than 17% of the values is given as follows:

X = 339.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 300, \sigma = 41[/tex]

The value of X that is less than 17% of the measures is X when Z has a p-value of 0.83, so X when Z = 0.955, hence:

0.955 = (X - 300)/41

X - 300 = 0.955 x 41

X = 339.

The problem asks for the value of X that is less than 17% of the values.

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On solving the linear equations x + y = 8 and x + 2y = 10, the value for peaches and mangoes is 6 and 2 respectively.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

Let the number of peaches be = x

Let the number of mangoes be = y

Total number of peaches and mangoes Morgan bought = 8

The linear equation for Morgan if she has bought 8 fruits -

x + y = 8 ....(1)

The cost of one peach is = $1

The cost of one mango is = $2

Total amount for Morgan to spend is = $ 10

The linear equation for Morgan if she has $10 to spend -

x + 2y = 10 ....(2)

Subtracting equation (1) from (2) -

x + 2y - (x + y) = 10 - 8

x + 2y - x - y = 2

y = 2

Substituting the value of y in equation (1) -

x + y = 8

x + 2 = 8

x = 8 - 2

x = 6

Now, to check equate the values of x and y in equation (2) -

x + 2y = 10

6 + 2(2) = 10

6 + 4 = 10

10 = 10

The graph is also plotted and it can be seen that the equations intersect at point (6,2).

Therefore, Morgan bought x = 6 peaches and y = 2 mangoes.

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The value of the function for f(5) is 133

A function can be defined as an equation, law, rule or expression that explains the relationship between two variables.

These variables are known as;

The different types of functions in mathematics are;

From the information given, we have that;

f(x) = 4x² + 2x - 2

To determine the function, f(5), substitute the value of x as 5

f(5) = 4(5)² + 2(5) -2

expand the bracket

f(5) = 125 + 10 -2

Add or subtract the values

f(5) = 133

Hence, the value is 133

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Answer:

g(f(- 5)) = - 31

Step-by-step explanation:

to calculate g(f(- 5)), evaluate f(- 5) and substitute the value obtained into g(x)

f(- 5) = (- 5)² + (- 5) - 12 = 25 - 5 - 12 = 25 - 17 = 8 , then

g(8) = - 2(8) - 15 = - 16 - 15 = - 31

Answer:

 -31

Step-by-step explanation:

f(x)=x²+x-12

or,f(-5)=(-5)²+(-5)-12

 =25-5-12

 =25-17

 =8

now, g(x)= -2x-15

 = (-2)×8-15

 = -16-15

 =-31

The value of a sub n, we must start with the previous value of a sub n minus 1, and then add four to it.

a_n = a_n-1 + 4

This formula is used to represent the recursive definition of a subscript n baseline. The formula is read as "a sub n is equal to a sub n minus 1 plus 4". This means that in order to find the value of a sub n, we must start with the previous value of a sub n minus 1, and then add four to it.

For example, let's say we want to find the value of a sub 5. We can start by looking at the previous value of a sub 4, which is 12. We then add four to 12, giving us 16. Therefore, a sub 5 is equal to 16.

To calculate this in a more general way, we can say that a sub n is equal to 4 times n minus 1, plus the initial value of a sub 1. That is, a sub n is equal to 4 (n-1) + 4. For example, a sub 5 is equal to 4 (5-1) + 4, which is equal to 16.

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