Find the volume of the solid whose base is the region inside the circle x² + y² = 9 if cross sections taken perpendicular to the y‐axis are squares.

The perimeter of triangle ABC with point A(-2,1) , B(-6,-3) and C(-4,4)  is 4√2 + √53 + √13

To find the perimeter of triangle ABC, we need to find the distance between its three vertices.

Using the distance formula, we can find the length of each side of the triangle.

AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-6 - (-2))² + (-3 - 1)²]

= √[(-4)² + (-4)²]

= √32

= 4√2

BC = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-4 - (-6))² + (4 - (-3))²]

= √[2² + 7²]

= √53

AC = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-4 - (-2))² + (4 - 1)²]

= √[2² + 3²]

= √13

Therefore, the perimeter of triangle ABC is:

Perimeter = AB + BC + AC

= 4√2 + √53 + √13

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The shape formed is rectangle. Length of AD is 5. Perimeter of ABCD is 20. Area of BCD is  [tex]\sqrt{65}[/tex] .

a) The shape formed is a rectangle.

b) The length of AD can be found using the distance formula:

AD = [tex]\sqrt{(3-(-2))^{2}+(-3-(-3))^{2} }[/tex]

     = [tex]\sqrt{5^{2}}[/tex]

     = 5

Therefore, the length of AD is 5.

c) The perimeter of ABCD can be found by adding up the lengths of all four sides:

AB = [tex]\sqrt{(-2-(-2))^{2}+(2-(-3))^{2} }[/tex]

     = [tex]\sqrt{5^{2}}[/tex]

     = 5

BC = [tex]\sqrt{(3-(-2))^{2}+(2-2)^{2} }[/tex]

     = [tex]\sqrt{5^{2}}[/tex]

     = 5

CD = [tex]\sqrt{(3-3)^{2}+(-3-2)^{2} }[/tex]

     = [tex]\sqrt{5^{2}}[/tex]

     = 5

DA = [tex]\sqrt{(-2-3)^{2}+(-3-(-3))^{2} }[/tex]

      = [tex]\sqrt{5^{2}}[/tex]

       = 5

Perimeter = AB + BC + CD + DA

                 = 5 + 5 + 5 + 5

                 = 20

Therefore, the perimeter of ABCD is 20.

d) Now join points B and D to form line segment BD. The area of triangle BCD can be found using the formula for the area of a triangle:

Area of BCD = (1/2) * base * height

The base is BD, which has length:

BD = [tex]\sqrt{(3-(-2))^{2}+(-3-2)^{2} }[/tex]

     = [tex]\sqrt{65}[/tex]

To find the height, we need to draw a perpendicular line from C to line BD:

The height is the length of the perpendicular line from C to line BD. Since C and D have the same x-coordinate, this perpendicular line will be vertical and have length 2 units (the difference between the y-coordinates of C and D).

Therefore, the height is 2.

Area of BCD = (1/2) * BD * height

                     = (1/2) *  [tex]\sqrt{65}[/tex] * 2

                     =  [tex]\sqrt{65}[/tex]

Therefore, the area of BCD is  [tex]\sqrt{65}[/tex] square units.

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The model of for the given relationship is,

R(x) = (200 + 50x)*(10 - x), where R(x) is the revenue of one week

This graph has downward parabola, vertex at (3, 2450) intersects X axis at 10 and Y axis at 40.

Hence the Graph Y.

Given that a candlemaker prices one set of scented candles at $10 and sells an average of 200 sets each week.

If he reduces the price by $1 then the sells increases 50 more per week.

When he will reduce $ x then the sells will increase 50x per week.

Now the price of each set of scented candles = (10 - x)

and sells in a week = (200 + 50x)

So if the candlemaker's weekly revenue is R(x) then

R(x) = (200 + 50x)*(10 - x)

R(x) = 2000 + 500x - 200x - 50x²

R(x) = 2000 + 300x - 50x²

If R(x) = y, then

y = 2000 + 300x - 50x²

50(x² - 6x - 40) = - y

50{(x - 3)² - 49} = - y

50(x - 3)² - 2450 = - y

50(x - 3)² = - (y - 2450)

So, the vertex at (3, 2450) and the parabola is downwards.

when intersect X axis then y = 0

x² - 6x - 40 = 0

x² - 10x + 4x - 40 = 0

(x - 10)(x + 4) = 0

x = -4, 10

and where cuts Y axis then x = 0

y = 40

Hence the correct graph is Graph Y.

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The integral that gives the length of the parametric curve x(t) = 2t², y(t) = t, 0 ≤ t ≤ 8 is: ∫₀⁸ √(16t² + 1) dt

To find the length of the parametric curve

[tex]x(t) = 2t^2, y(t) = t, 0 ≤ t ≤ 8[/tex]

, we need to evaluate the integral:

[tex]∫√[(dx/dt)² + (dy/dt)²]dt[/tex]

where dx/dt and dy/dt are the first derivatives of x(t) and y(t) with respect to t, respectively.

Step 1: Compute dx/dt ,

[tex] dx/dt = d(2t^2)/dt = 4t dy/dt = d(t)/dt =1[/tex]

Step 2: Compute the square of the derivatives and sum them

(4t)² + (1)² = 16t² + 1

Step 3: Find the square root of the sum √(16t² + 1) Step 4: Set up the integral and evaluate it over the given interval [0, 8] Length = ∫₀⁸ √(16t² + 1) dt

Thus, the integral that gives the length of the parametric curve x(t) = 2t², y(t) = t, 0 ≤ t ≤ 8 is: ∫₀⁸ √(16t² + 1) dt

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Correct answer is "Examine the following integral gives the length of the parametric curve or not x(t) = 2t², y(t) = t, 0 ≤t≤8"

According to the regression equation, we can estimate that a student with a test grade of 68 likely scored around 66 on their homework.

The linear regression equation is represented as:

y = mx + b

where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept. In this case, y represents the test grade, and x represents the homework grade.

To calculate the linear regression equation for this data set, we can use a statistical software or a calculator. The resulting equation for this data set is:

y = 0.85x + 12.06

This equation tells us that for every one-point increase in the homework grade (x), the test grade (y) increases by 0.85 points. The y-intercept of 12.06 tells us that if a student scored a 0 on their homework, they would still be expected to receive a 12.06 on their test.

Using this equation, we can estimate the homework grade for a student with a test grade of 68. To do this, we can plug in 68 for y and solve for x:

68 = 0.85x + 12.06

55.94 = 0.85x

x ≈ 65.81

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Complete Question:

A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, estimate the homework grade, to the nearest integer, for a student with a test grade of 68.

Homework Grade (x) Test Grade (y)

X | Y

88 | 90

55 | 55

89 | 91

85 | 88

61 | 52

76 | 76

76 | 81

61 | 59

Answer:

To convert degrees to radians, you can use the following formula:radians = (degrees x pi) / 180Using this formula, we can convert 70 degrees to radians as follows:radians = (70 x pi) / 180

radians = 1.2217304764Therefore, 70 degrees is equal to approximately 1.22 radians.

Step-by-step explanation:

[tex]\sf \dfrac{7\pi}{18}(rad).[/tex]

So a conversion factor is basically a fraction compounded by a numerator and denominator that are equivalent values of different units. With these factors you always want the resulting unit as a numerator and the unit to be cancelled at the bottom.

In the case of angle measurement, 2π radians is equivalent to 360° degrees. Therefore, the following conversion factor can be used when converting from degrees to radians:

[tex]\sf \dfrac{2\pi (rad)}{360(deg)}.[/tex]

We can also use the following to convert from radians to degrees:

[tex]\sf \dfrac{360(deg)}{2\pi (rad)}.[/tex]

Now, we just need to multiply our 70 degrees by the corresponding conversion factor:

[tex]\sf 70(deg)\dfrac{2\pi (rad)}{360(deg)}[/tex]

Let's  isolate "π" to give and answer in terms of "π".

[tex]\sf \dfrac{70(deg)(2) }{360(deg)}[(\pi)(rad)]=\\ \\\\ \dfrac{140(deg)}{360(deg)}[(\pi)(rad)]=\\ \\ \\\dfrac{7\pi}{18}(rad).[/tex]

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The equation that describes the translation is given as follows:

(x,y) -> (x - 4, y + 3).

The four translation rules are defined as follows:

If we look the corresponding vertex of each vertex, we have that the translations are given as follows:

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Answer:

QS = 5√3, so RS = 5√3√3 = 15 feet.

The correct answer is B.

After simplifying the expression 8⁻⁵ / 8⁻⁷ with no exponents, we get the result as 64.

To simplify 8⁻⁵ / 8⁻⁷ with no exponents, we need to use the rule that states when dividing two powers with the same base, you can subtract the exponents. Thus:

8⁻⁵ / 8⁻⁷ = 8⁻⁵ x 8⁷

To simplify this further, we can use the rule that states when multiplying powers with the same base, you can add the exponents. Thus:

8⁻⁵ x 8⁷ = 8⁻⁵⁺⁷

Simplifying the exponent by adding -5 and 7, we get:

8²

Therefore, 8⁻⁵ / 8⁻⁷ with no exponents is equal to 8², which simplifies to 64. So the final answer is 64.

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Answer:

the answer is B

Step-by-step explanation:

are you a make a wish kid

Because i would love for you to bounce that

For the first problem, using the monthly payment formula monthly deposit needed is $118.69. For the second problem, using the same formula the monthly payment is $1029.73.

We have the following variables

P = Monthly payment

r = Annual interest rate = 3.7% = 0.037/12 per month

n = Number of payments = 15 months

A = Amount to be saved = $1800

Using the monthly payment formula

P = (r * A) / (1 - (1 + r)⁻ⁿ)

Substituting the given values

P = (0.003083 * 1800) / (1 - (1 + 0.003083)⁻¹⁵)

P ≈ $118.69

Therefore, you should deposit approximately $118.69 each month to save $1800 in 15 months, assuming an APR of 3.7%, compounded monthly.

We have the following variables

P = Monthly payment

r = Annual interest rate = 9.1% = 0.091/12 per month

n = Number of payments = 30 years * 12 months = 360 months

A = Mortgage amount = $127,000

Using the monthly payment formula

P = (r * A) / (1 - (1 + r)⁻ⁿ)

Substituting the given values

P = (0.007583 * 127000) / (1 - (1 + 0.007583)⁻³⁶⁰)

P ≈ $1029.73

Therefore, the monthly payment for a shack mortgage of $127,000 with a fixed APR of 9.1% for 30 years would be approximately $1029.73.

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Distance = 75 units between t=0 and t=5. To find how far the particle travels between t=0 and t=5, we need to calculate the total distance traveled by the particle.

The distance traveled is equal to the total displacement, which can be found by taking the absolute value of the difference between the initial and final positions.

The initial position of the particle at t=0 is:

S(0) = 0^3 - 3(0)^2 - 2 = -2

The final position of the particle at t=5 is:

S(5) = 5^3 - 3(5)^2 - 2 = 73

Therefore, the total displacement of the particle is:

ΔS = |73 - (-2)| = 75

So the particle travels a total distance of 75 units between t=0 and t=5.

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A two-tailed test, if the test statistic (such as z) is far from 0, the p-value will be small (closer to 0).

Smaller p-value, which signifies a lower probability of observing such a test statistic under the null hypothesis.

Reject the null hypothesis and conclude that there is a significant difference between the parameters being tested.

A two-tailed test, if the test statistic (such as z) is far from 0, the p-value will be small (closer to 0).

Let's go through the process step-by-step:
Hypothesis:

In a two-tailed test, we consider two opposite hypotheses.

The null hypothesis (H0) states that there is no significant difference between the parameters being tested, and the alternative hypothesis (H1) states that there is a significant difference.
Test statistic:

The test statistic (e.g., z) is a standardized value that helps us compare our sample data to the expected population data.

The further the test statistic is from 0, the more it deviates from the null hypothesis.
p-value:

The p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated from the sample data, assuming the null hypothesis is true.
Decision:

In a two-tailed test, we compare the p-value to a predetermined significance level (usually 0.05).

If the p-value is less than the significance level, we reject the null hypothesis in favor of the alternative hypothesis.
The test statistic is far from 0, it indicates a greater deviation from the null hypothesis.

This results in a smaller p-value, which signifies a lower probability of observing such a test statistic under the null hypothesis.

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Based on the provided list, the four true statements regarding confidence intervals for a population mean are:

1. If a confidence interval for the population mean is constructed from a sample of size n=30, that interval must contain the sample mean.

2. If a particular 91% confidence interval captures the population mean, then for the same sample data, the population mean will also be captured at the 87% confidence level.

3. Increasing the confidence level will increase the width of the confidence interval.

4. For the same sample data, a 95% confidence interval will be narrower than a 99% confidence interval.

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The scientific notation 3.05E-8 represents 3.05 × 10⁻⁸, and its equivalent is 0.0000000305. Option B

Scientific notation is a way of expressing very large or very small numbers in a more concise and clearer form.

A number in scientific notation is written as the product of two factors which are, a coefficient and a power of 10. It comes in the form a × 10ᵇ

'a' is called the coefficient and 'b' is the exponent of 10.

For example, the number 3,000,000 can be written in scientific notation as 3 × 10⁶ and the number 0.00000045 can be written as 4.5 × 10⁻⁷

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An equation that could be used to find p, the base price of the pizza is: D. 7.15 = 1/3(p + 4.5).

In order to write a linear equation to describe this situation, we would assign a variable to the base price of the pizza, and then translate the word problem into a linear equation as follows:

Let the variable p represent the base price of the pizza.

Since this group of three (3) friends had a share of $7.15, which included the base price of the pizza, a linear equation that can be used to model this situation is given by;

7.15 = 1/3(p + 4.5)

21.45 = p + 4.5

p = 21.45 - 4.5

p = $16.95

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If the probability that Mary will win a game is 0.03, and loosing game is 0.97 then the excepted value of game is equals to -$1.11.

Expected Value of the game is the mean of the probability distribution of the payout values, denoted by E(X). It is equal to the sum of the products of each possible payout value and its corresponding probability, that is[tex]E( x) = \sum_{i } x_i p( x_i) \\[/tex]

where, xᵢ --> payouts

p(xᵢ) --> probability for corresponding to payouts. Let's consider X be a variable denotes the payouts ( the amount the player wins for a particular outcome of the game). Here possible value of x are $60 and -$3. Now, determine probabilities corresponding to payouts.

Probability that Mary will win a game= 0.03

Probability that marry will not win a game

= 1 - 0.03 = 0.97

So, Probability distribution table is

x $60 -$3

P(x) 0.03 0.97

Now, using formula of excepted value,

E(X) = - (prize for winning game × (Probability of winning) + (Prize for loosing game)×(Probability of loosing the game or [tex]E( x)= \sum_{i } x_i p( x_i)\\ [/tex] Substitute all known values

= $60× 0.03 - $3× 0.97

= $1.80 - $2.91

= -$1.11

Hence required value is -$1.11.

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Therefore, the expression "four more than 5 times the cube of a number" is equivalent to 5x³ + 4.

Let the number be represented by x. Then, "5 times the cube of a number" can be written as 5x³. "Four more than 5 times the cube of a number" means we add 4 to this expression.

The phrase "the cube of a number" means that we raise the number to the third power, which is denoted by x³.

The phrase "5 times the cube of a number" means that we multiply the cube of the number by 5, which is denoted by 5x³.

The phrase "four more than 5 times the cube of a number" means we add 4 to 5x³, which is denoted by 5x³ + 4.

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The smallest number of words which could be there in the set is equal to 56.

The smallest number of words that could be in the set,

Find the least common multiple LCM of 8 and 14,

Since that will be the smallest number that is divisible by both 8 and 14.

The prime factorization of 8 is 2 × 2 × 2,

while the prime factorization of 14 is 2 × 7.

To find the least common multiple LCM,

Take the highest power of each prime factor that appears in either factorization and multiply them together.

Thus we have,

LCM(8, 14) = 2 × 2 × 2 × 7

⇒ LCM(8, 14)= 56

Therefore, the smallest number of words in the set could be 56.

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The alternative hypothesis of the chi-square test is that the obtained chi-square value will be significantly different from the expected chi-square value under the null hypothesis

The alternative hypothesis of the chi-square test states that the obtained chi-square value will be significantly different from the expected chi-square value under the null hypothesis. In other words, the alternative hypothesis states that there is a relationship between the variables being tested, and that the observed data is not due to chance alone. The null hypothesis, on the other hand, asserts that there is no correlation between the variables under consideration and that any differences that are seen are the result of chance.

Therefore, the alternative hypothesis of the chi-square test is that the obtained chi-square value will be significantly different from the expected chi-square value under the null hypothesis, indicating that there is a relationship between the variables being tested.

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As we have proven both directions of the statement X = Y if and only if Xy = (X + Y)²/4.

Let's start with the "if" direction. This means we need to prove that if Xy = (X + Y)²/4, then X = Y. To do this, we can start by multiplying both sides of the equation by 4 to get rid of the fraction:

4Xy = (X + Y)²

Expanding the right-hand side of the equation gives:

4Xy = X² + 2XY + Y²

We can rearrange this equation by subtracting 2XY and Y² from both sides:

4Xy - 2XY - Y² = X²

Next, we can factor out X on the left-hand side:

X(4y - 2Y) = X² - Y²

If we assume X ≠ 0, we can divide both sides by X to get:

4y - 2Y = X - Y

Simplifying this expression gives:

2y = X + Y

Finally, we can substitute this equation back into the original equation Xy = (X + Y)²/4 to get:

Xy = (2y)²/4

Simplifying this expression gives:

Xy = y²

Since X ≠ 0 (as we assumed earlier), we can divide both sides by X to get:

y = X

Therefore, we have shown that if Xy = (X + Y)²/4, then X = Y.

Now, let's move on to the "only if" direction. This means we need to prove that if X = Y, then Xy = (X + Y)²/4. To do this, we can start with the equation X = Y and substitute Y for X in the equation Xy = (X + Y)²/4:

Yy = (Y + Y)²/4

Simplifying this expression gives:

Yy = Y²

Dividing both sides by Y (since Y ≠ 0), we get:

y = Y/1

Therefore, we have shown that if X = Y, then Xy = (X + Y)²/4.

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Answer:

0.27

Step-by-step explanation:

Probability of picking a green is 0.27 (t nearest hundredth).

The given matrix is:

```

4  6  2

1  5 -3

-7 0  9

```

The dimensions of the matrix are 3 x 3 because it has 3 rows and 3 columns.

The indicated element is "a23", which is not present in the matrix. The correct element in the 2nd row, 3rd column is 9.

Therefore, the correct answer is (b) 3x3; 9, a23.

Step-by-step explanation:

- The matrix has 3 rows and 3 columns, so its dimensions are 3 x 3.

- The indicated element is a23, which does not exist in the matrix. However, the element in the 2nd row and 3rd column is 9.

- Therefore, the correct answer is (b) 3x3; 9, a23.

The three-decimals which are written in expanded form and are greater than 5.3 are (b) 5+0.3+0.001,  (d) (5×1)+(3×0.1)+(3×0.01)  and (e) (6×1)+(2×0.1)+(1×0.01).

A Decimal is a number system that uses a base of ten and a decimal point to separate the whole number from the fractional part. It represents numbers that are not whole or integers, but rather parts of a whole.

To determine which decimals written in expanded-form are greater than 5.3, we simply add up the values of each place value.

The decimals that are greater than 5.3 will have a sum greater than 5.3.

Option (a) : 5+0.3+0.00

= 5 + 0.3,

= 5.3, which is not-greater than 5.3.

Option (b) : 5+0.3+0.001

= 5.3 + 0.001,

= 5.301, which is greater than 5.3.

Option (c) : (5×1)+(2×0.1)+(9×0.01)

= 5 + 0.2 + 0.09,

= 5.2 + 0.09,

= 5.29, which is not greater than 5.3.

Option (d) : (5×1)+(3×0.1)+(3×0.01)

= 5 + 0.3 + 0.03,

= 5.3 + 0.03,

= 5.33, which is greater than 5.3.

Option (e) : (6×1)+(2×0.1)+(1×0.01)

= 6 + 0.2 + 0.01,

= 6.2 + 0.01

= 6.21, which is greater than 5.3.

Therefore, the decimals that are greater than 5.3 are options (b), (d), and (e).

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The given question is incomplete, the complete question is

Which decimals written in expanded form are greater than 5.3 ?

Select three options.

(a) 5+0.3+0.00

(b) 5+0.3+0.001

(c) (5×1)+(2×0.1)+(9×0.01)

(d) (5×1)+(3×0.1)+(3×0.01)

(e) (6×1)+(2×0.1)+(1×0.01)

The price of the used book in the bookstore is $7.50.

A price is amount of money that the buyer pays to seller in an exchange for a product, service, asset etc.

We are given that:

The owner of the bookstore buys a used book for $2.50.

He resells it for 300% of that price.

So, the price of the used book in this bookstore will be:

= $2.50 x 300%

= $2.50 x 3.00

= $7.50.

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The unknown values in the line segment area s follows:

y = 9

GH = 66 units

HI = 22 units

GI =  88 units

The line segment is measured GI. Therefore, H is in between G and I.

Hence,

GH = 7y + 3

HI = 3y - 5

GI = 9y + 7

Let's find y as follows:

7y + 3 + 3y - 5 = 9y + 7

10y - 2 = 9y + 7

10y - 9y = 7 + 2

y = 9

Let's find the GH

GH = 7y + 3 = 7(9) + 3= 63 + 3 = 66 units

Let's find the HI

HI = 3y - 5 = 3(9) - 5 = 27 - 5 = 22 units

Let's find the GI

GI = 9y + 7 = 9(9) + 7 = 81 + 7 = 88 units

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a. The random variable X is the number of California residents in a sample of 11 who have adequate earthquake supplies.

b. X may take on the values 0, 1, 2, ..., 11, since it is possible that none or all of the residents have adequate earthquake supplies.

c. X ~ Binomial(11, 0.3), since we have a fixed number of trials (n=11) and each trial is either a success (having adequate earthquake supplies) or a failure (not having adequate earthquake supplies), and the probability of success (p=0.3) is constant for each trial.

d. To find the probability that at least eight have adequate earthquake supplies, we can use the binomial probability formula or a calculator. Using a calculator, we get P(X >= 8) = 0.0512.

e. It is more likely that none of the residents surveyed will have adequate earthquake supplies.

f. The expected value of X is E(X) = np = 11 x 0.3 = 3.3. So we can expect about 3 or 4 residents out of the 11 to have adequate earthquake supplies on average.

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

a. The random variable X is the number of California residents in a sample of 11 who have adequate earthquake supplies.

b. X may take on the values 0, 1, 2, ..., 11, since it is possible that none or all of the residents have adequate earthquake supplies.

c. X ~ Binomial(11, 0.3), since we have a fixed number of trials (n=11) and each trial is either a success (having adequate earthquake supplies) or a failure (not having adequate earthquake supplies), and the probability of success (p=0.3) is constant for each trial.

d. To find the probability that at least eight have adequate earthquake supplies, we can use the binomial probability formula or a calculator. Using a calculator, we get P(X >= 8) = 0.0512.

e. It is more likely that none of the residents surveyed will have adequate earthquake supplies. This is because the probability of success (having adequate earthquake supplies) for each resident is only 0.3, which means the probability of all 11 residents having adequate supplies is very low [tex]0.3^{11}[/tex], while the probability of none of them having adequate supplies is much higher [tex](0.7^{11})[/tex].

f. The expected value of X is E(X) = np = 11 x 0.3 = 3.3. So we can expect about 3 or 4 residents out of the 11 to have adequate earthquake supplies on average.

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It will cost $211.05 to pour the concrete slab for the patio.

First, we have to convert the dimensions of the patio into yards.

11 feet = 3.67 yards (since there are 3 feet in a yard)

Next, we need to convert the depth of the concrete from inches to yards.

3 inches = 0.25 yards (since there are 36 inches in a yard)

Volume of patio is

Volume = Length x Width x Depth

= 3.67 yards x 3.67 yards x 0.25 yards

= 3.35 cubic yards

Cost = Volume x Price

= 3.35 cubic yards x $63.00

= $211.05

Therefore, it will cost $211.05 to pour the concrete slab for the patio.

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The probability that two telephone calls come into a switchboard within three minutes of each other in a one-hour period is approximately 0.6 or 60%. This is calculated using the uniform distribution and the probability density function.

To solve this problem, we can assume that the first call comes at a random time, and we need to find the probability that the second call comes within three minutes of the first call.

Let's assume that the first call comes at time t, where t is a random number between 0 and 60 minutes. Then the probability that the second call comes within three minutes of the first call is the probability that the second call comes between t-3 and t+3 minutes.

Since the second call is also a random event, we can assume that it has an equal probability of occurring at any time during the one-hour period. Therefore, the probability that the second call comes between t-3 and t+3 minutes is

P(t-3 < second call < t+3) = (t+3 - (t-3))/60 = 6/60 = 1/10

This probability holds for any value of t between 0 and 60. Therefore, we need to integrate this probability over the entire range of possible values of t

P(calls within 3 minutes of each other) = ∫(0 to 60) P(t-3 < second call < t+3) dt

= [tex]\int\limits^0_{60}[/tex] (1/10) dt

= (1/10) [tex]\int\limits^0_{60}[/tex] dt

= (1/10) [t] from 0 to 60

= (1/10) (60)

= 6/10

= 0.6

Therefore, the probability that the calls are made within three minutes of each other is 0.6 or 60%.

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--The given question is incomplete, the complete question is given

" Two telephone calls come into a switchboard at random times in a fixed one-hour period. Assume that the calls are made independently of one another. What is the probability that the calls are made within three minutes of each other? "--