A square pyramid has a height of 8 inches and a volume of 600 cubic inches. What is the area of the base of the pyramid?

As per the given equation, the value of c guaranteed by the Mean Value Theorem for the function f(x) = √(9 - [tex]x^2[/tex]) over the interval [0, 3] is c = ± √(9/5).

To find the value of c guaranteed by the Mean Value Theorem for the function f(x) = √(9 - [tex]x^2[/tex]) over the interval [0, 3], we need to determine if the conditions of the Mean Value Theorem are satisfied and then find the value of c.

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one value c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

In our case, the function f(x) = √(9 - [tex]x^2[/tex]) is continuous on the closed interval [0, 3] since it is a square root function and the radicand is always non-negative within this interval.

The function is also differentiable on the open interval (0, 3) since it is the square root of a differentiable function.

To find the value of c, we first calculate f(3) and f(0):

f(3) = √(9 - [tex]3^2[/tex]) = √(9 - 9) = √0 = 0

f(0) = √(9 - [tex]0^2[/tex]) = √(9 - 0) = √9 = 3

Next, we calculate f'(c):

f'(x) = (-2x)/√(9 - x^2)

We want to find the value of c such that f'(c) = (f(3) - f(0))/(3 - 0). Let's substitute the values into the equation:

(-2c)/√(9 - [tex]c^2[/tex]) = (0 - 3)/(3 - 0)

(-2c)/√(9 - [tex]c^2[/tex]) = -1

To solve for c, we can cross-multiply:

-2c = -√(9 - [tex]c^2[/tex])

Squaring both sides:

4c^2 = 9 - [tex]c^2[/tex]

Simplifying:

5[tex]c^2[/tex] = 9

Dividing both sides by 5:

[tex]c^2[/tex] = 9/5

Taking the square root of both sides:

c = ± √(9/5)

Therefore, the value of c guaranteed by the Mean Value Theorem for the function f(x) = √(9 - [tex]x^2[/tex]) over the interval [0, 3] is c = ± √(9/5).

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The graph of the function y=cos(x) is shown below:

[Image of the graph of the function y=cos(x)]

The graph of the function y=cos(2x) is shown below:

[Image of the graph of the function y=cos(2x)]

The graph of the function y=cos(4x) is shown below:

[Image of the graph of the function y=cos(4x)]

As you can see, the graph of the function y=cos(4x) is the same as the graph of the function y=cos(x), but it is compressed horizontally by a factor of 4. This is because the period of the function y=cos(4x) is 2π/4=π/2, which is half the period of the function y=cos(x).

Therefore, the correct answer is A.

Answer:

d

Step-by-step explanation:

Answer:

y=8

Step-by-step explanation:

The width, w, of the opening between x and y, is 6.9 ft.

We have,

From the diagram,

We have the radius of the circle and the angle subtended by the chord at the center of the circle.

So,

We can also use the formula.

= 2 x radius x sin(angle/2)

This is the length of the chord.

Now,

radius = 4 ft

angle = 360/3 = 120

Substituting the values.

= 2 x radius x sin(angle/2)

= 2 x 4 x sin (120/2)

= 2 x 4 x sin 60

= 2 x 4 x √3/2

= 4 x √3

= 4 x 1.732

= 6.928

Rounding to the nearest tenth.

= 6.9 ft

Thus,

The width, w, of the opening between x and y, is 6.9 ft.

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Answer: Inflation is the rise in general level of prices which is determined by the Consumer Price Index (CPI).

Answer:

The x-intercepts are (-3, 0) and (2, 0).

Answer:

Expected winnings = $1.33

Expected profit or loss = -$2.67

Step-by-step explanation:

Winning Probability Prize Value

Yes   1/150         $200

No         149/150         $0

Expected winnings of one raffle ticket:

Expected winnings = (Probability of winning x Prize value) + (Probability of losing x Prize value)

Expected winnings = (1/150 x $200) + (149/150 x $0)

Expected winnings = $1.33

If a raffle ticket costs $4, the expected profit or loss of one raffle ticket can be calculated as:

Expected profit or loss = Expected winnings - Cost of ticket

Expected profit or loss = $1.33 - $4

Expected profit or loss = -$2.67

hello

to answer precisely, i have to see the graphs in the question but overall, the answer should look like something like the graph in the attached file

1)

The measure of angle A is 45° .

Given,

a= 60

b = 60√2

c = 60

Now the given sides with the dimension is of right angled triangle.

So,

Angle B = 90°

Angle A = 45°

Angle C = 45°

2)

All the six trigonometric functions:

SinФ = 60/60√2

cosФ = 60/60√2

tanФ = 60/60

cosecФ = 60√2/60

secФ = 60√2 /60

cotФ = 60/60

3)

Area of triangle = 1/2 × b× h

Area of triangle = 1/2 *60*60

Area of triangle = 1800 in²

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The function g(x) is increasing exponentially because it eventually exceeds f(x).

The table shows that as x increases, the values of g(x) increase at a much faster rate compared to the values of f(x).

In fact, the values of g(x) are growing exponentially, as each subsequent value is significantly larger than the previous one.

Therefore, the function g(x) is most likely increasing exponentially.

Thus, g(x) is increasing exponentially because it eventually exceeds f(x).

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The value of the side labelled x is equal to 3.8 to the nearest tenth using the trigonometric ratio of sine.

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

Considering the sine of angle 41°

sin 41° = 2.5/x {opposite/hypotenuse}

x = 2.5/sin 41° {cross multiplication}

x = 3.8106

Therefore, the value of the side labelled x is equal to 3.8 to the nearest tenth using the trigonometric ratio of sine.

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The stem display for the data, given the monthly rent and the apartment building is:

$14 | 10 15 35

$15 | 15 55 75 95

$16 | 40 90

To create a stem display, we first divide the data into groups of tens. The first group is $ 1400  -$ 1499, the second group is  $ 1500 - $1599 , and so on. Next, we write the units digit of each number in the appropriate group.

For example, the number 1,710 is in the $ 1500 - $ 1599 group, so we write a 1 in the 15 column. Finally, we count the number of numbers in each group. There are three numbers in the $ 1400 - $ 1499 group, two numbers in the $ 1500 - $ 1599 group, and three numbers in the $ 1600 - $ 1699 group.

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The cubic model that fits the given data is: y ≈ 0.00006204x³ + 0.00231059x² - 0.10811757x + 18.2

Let's set up a system of equations using the given data points:

For x = 0 (year 1950):

18.2 = a(0)³ + b(0)² + c(0) + d ==> d = 18.2

For x = 10 (year 2000):

65.5 = a(10)³ + b(10)² + c(10) + 18.2

For x = 20 (year 2010):

75.3 = a(20)³ + b(20)² + c(20) + 18.2

For x = 25 (year 2015):

78.1 = a(25)³ + b(25)² + c(25) + 18.2

For x = 30 (year 2020):

78.5 = a(30)³ + b(30)² + c(30) + 18.2

For x = 40 (year 2030):

80.5 = a(40)³ + b(40)² + c(40) + 18.2

For x = 50 (year 2040):

86.6 = a(50)³ + b(50)² + c(50) + 18.2

For x = 60 (year 2050):

91.5 = a(60)³ + b(60)² + c(60) + 18.2

Simplifying these equations, we have: d = 18.2

1000a + 100b + 10c + 18.2 = 65.5

8000a + 400b + 20c + 18.2 = 75.3

15625a + 625b + 25c + 18.2 = 78.1

27000a + 900b + 30c + 18.2 = 78.5

64000a + 1600b + 40c + 18.2 = 80.5

125000a + 2500b + 50c + 18.2 = 86.6

216000a + 3600b + 60c + 18.2 = 91.5

We can solve this system of equations to find the coefficients a, b, and c.

Using a cubic regression calculator or a matrix solver, the solution is:

a ≈ 0.00006204

b ≈ 0.00231059

c ≈ -0.10811757

Therefore, the cubic model that fits the given data is: y ≈ 0.00006204x³ + 0.00231059x² - 0.10811757x + 18.2

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Answer: y = 550x

Step-by-step explanation:

Answer:

  125/384 ≈ 0.32552

Step-by-step explanation:

You want the area between the curves y = 5x and y = 8x².

The difference between the curves is ...

  f(x) = 5x -8x² = x(5 -8x)

This difference is zero when ...

  x = 0

  5 -8x = 0   ⇒   x = 5/8

The area will be the integral of f(x) with the limits 0 and 5/8:

  [tex]\displaystyle \text{area}=\int_0^\frac{5}{8}{(5x-8x^2)}\,dx=\dfrac{5}{2}\cdot\left(\dfrac{5}{8}\right)^2-\dfrac{8}{3}\cdot\left(\dfrac{5}{8}\right)^3\\\\\\\text{area}=\left(\dfrac{5}{8}\right)^2\left(\dfrac{5}{2}-\dfrac{8}{3}\cdot\dfrac{5}{8}\right)=\dfrac{25}{64}\cdot\dfrac{5}{6}=\boxed{\dfrac{125}{384}}[/tex]

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a) The mean of the random variable is 2.610 (rounded to 2 decimal places).

b) The variance of the random variable is 1.880 (rounded to 3 decimal places).

c) The standard deviation of the random variable is 1.372 (rounded to 3 decimal places).

d) The probability that a randomly selected student has a parent involved in 4 activities is 0.260 (rounded to 3 decimal places).

To compute the variance of the random variable, we need to calculate the squared deviation of each value from the mean, weighted by their respective probabilities, and then sum them up.

b) Variance [tex](\sigma^2)[/tex] of the random variable:

Variance is given by the formula[tex]Var(X) = \sum [(X - \mu)^2 \times P(X)],[/tex] where X represents the values of the random variable, μ is the mean, and P(X) is the probability.

Using the given data:

X: 0 1 2 3 4

P(X): 0.053 0.117 0.258 0.312 0.26

μ (mean): 2.610

Calculating the squared deviations for each value:

[tex](0 - 2.610)^2 \times 0.053 = 14.152[/tex]

[tex](1 - 2.610)^2 \times 0.117 = 0.291[/tex]

[tex](2 - 2.610)^2 \times 0.258 = 0.148[/tex]

[tex](3 - 2.610)^2 \times 0.312 = 0.122[/tex]

[tex](4 - 2.610)^2 \times 0.26 = 1.429[/tex]

Summing up the squared deviations:

Var(X) = 14.152 + 0.291 + 0.148 + 0.122 + 1.429 = 16.142

Therefore, the variance of the random variable is 16.142.

c) Standard deviation (σ) of the random variable:

The standard deviation is the square root of the variance.

Taking the square root of the variance calculated above:

Standard deviation (σ) = √(16.142) ≈ 4.020 (rounded to 3 decimal places)

d) Probability of a randomly selected student having a parent involved in 4 activities:

The probability of a specific value occurring in a discrete probability distribution is given by the corresponding probability value.

From the given data:

P(X = 4) = 0.26

Therefore, the probability that a randomly selected student has a parent involved in 4 activities is 0.26 (rounded to 3 decimal places).

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Answer:

I love math

Step-by-step explanation:

The formula for simple interest is:

I = P * r * t

where:

I = interest earned

P = principal amount

r = interest rate (as a decimal)

t = time (in years)

In this case, we know that:

P = $1,264

r = 2.75% = 0.0275 (as a decimal)

t = 2.5 years

So, we can plug in these values and solve for I:

I = 1,264 * 0.0275 * 2.5 = $87.55

Therefore, the interest earned over 2.5 years is $87.55. To find the ending balance, we need to add the interest earned to the principal:

Ending balance = $1,264 + $87.55 = $1,351.55

So, Quentin's account balance will be $1,351.55 at the end of 2.5 years.

The inequality that represents the number sentence is given as follows:

w + 4.6 > 6.

The four most common inequality symbols, and how to interpret them, are presented as follows:

A number w plus 4.6 is represented as follows:

w + 4.6.

The amount is more than 6, hence the inequality is given as follows:

w + 4.6 > 6.

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Answer:

Kendra's Tea Shop served 87 caffeinated teas and 13 decaffeinated teas. This means that 87 out of 100 teas served were caffeinated. Therefore, 87% of the teas served were caffeinated.

The calculated area of the triangle is 38.30 square units

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

The triangle

The area of the triangle is calculated as

Area = 1/2absin(c)

Where

a = b = 10

c = 50 degrees

Using the above as a guide, we have the following:

Area = 1/2 * 10 * 10 * sin(50 degrees)

Evaluate

Area = 38.30

Hence, the area of the triangle is 38.30 square units

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The rate of the decay of the given function is 75 percent.

The given function is y=12,500(3/4)ˣ

This function represents exponential decay with a base of (3/4). The rate of decay, "r", is equal to 3/4. This means that the value of the function decreases by a factor of 3/4 every time the independent variable increases by 1. Thus, the rate of decay is 3/4 or 75%.

Therefore, the rate of the decay of the given function is 75 percent.

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Answer:

V = (4/3) × 3.14 × 14³ = 11,310.08 cubic feet

Work Shown:

d = diameter = 28

r = radius = d/2 = 28/2 = 14

V = volume of a sphere

V = (4/3)*pi*r^3

V = (4/3)*3.14*(14)^3

V = 11,488.2133 cubic feet approximately

The info "weighs almost 8 tons" is never used. It is likely an intended distraction. All we need is the radius of the sphere to get its volume.

Answer: if it's true that 10 > 7 (P), then it's also true that 10 > 5 (Q).

Step-by-step explanation: In the context of logic and truth tables, p → q can be read as "if p then q." You've provided the truth values for the combinations of p and q, which I'll summarize here:

If p and q are both False (F), then p → q is True (T).

If p is True (T) and q is False (F), then p → q is False (F).

If p is False (F) and q is True (T), then p → q is True (T).

If p and q are both True (T), then p → q is True (T).

Given your propositions:

P: 10 > 7

Q: 10 > 5

P is True because 10 is indeed greater than 7. Q is also True because 10 is greater than 5.

Therefore, we're in the fourth case of your truth table: both p and q are True, so p → q is also True.

The consumer surplus for the price of the dog food is given as follows:

$1.

The consumer surplus is defined as the difference between the amount a consumer is willing to pay for a product and the actual price they pay.

The parameters in the context of this problem are given as follows:

Hence the consumer surplus for the price of the dog food is given as follows:

5 - 4 = $1.

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Answer:

[tex]\huge\boxed{\sf x = 5}[/tex]

Step-by-step explanation:

20x + 30 = 28x - 10 (Corresponding angles)

20x + 30 + 10 = 28x

20x + 40 = 28x

40 = 28x - 20x

40 = 8x

5 = x

OR

[tex]\rule[225]{225}{2}[/tex]

Answer:

y = 3x + 3

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2} -x_{1} }[/tex]

with (x₁, y₁ ) = (- 1, 0) and (x₂, y₂ ) = (0, 3) ← 2 points on the line

m = [tex]\frac{3-0}{0-(-1)}[/tex] = [tex]\frac{3}{0+1}[/tex] = [tex]\frac{3}{1}[/tex] = 3

the line crosses the y- axis at (0, 3 ) ⇒ c = 3

y = 3x + 3 ← equation of line

Answer:

Step-by-step explanation:

Median:  List numbers from smallest to largest.  Median is middle number

2       2        3         5        6          8           10

                               |

5 is in the middle

Median = 5

The number that belongs in the green box is given as follows:

D.  2.

A quadratic function is given according to the following rule:

y = ax² + bx + c

The solutions are given as follows:

In which the discriminant is given as follows:

[tex]\Delta = b^2 - 4ac[/tex]

The number that goes into the green box is then given as follows:

2a = 2(1) = 2.

Meaning that option D is the correct option in the context of this problem.

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