Toby created a sculpture for art class using different-sized cubes. the smallest cube is 1.5 inches along each edge. the largest cube is 7.5 inches along each edge. how many of the smallest cubes would it take to fill the largest cube

Answer:

324π

Step-by-step explanation:

Area of circle = r² · π

r = 18 in

Find its area in terms of pi.

We Take

18² · π = 324π

So, the area of the circle is 324π.

Answer:

A = 324π

Step-by-step explanation:

A = πr²

A = π(18)²

A = π(324)

A = 324π

The maximum profit is $1,892,250.

The profit function is given by[tex]P(x) = -x^2 + 15x + 72x - 300 = -x^2 + 87x - 300.[/tex]

To find the number of hundred thousands of tires that must be sold to maximize profit, we need to find the value of x that maximizes P(x).

We can do this by finding the critical point of P(x) and then checking whether it corresponds to a maximum or a minimum.

The derivative of P(x) is:

P'(x) = -2x + 87

Setting this equal to zero to find critical points, we get:

-2x + 87 = 0

Solving for x, we get:

x = 43.5

Since the problem specifies that x must be at least 5, we know that the critical point x = 43.5 corresponds to a maximum.

Therefore, the number of hundred thousands of tires that must be sold to maximize profit is 43.5.

To find the maximum profit, we substitute x = 43.5 into the profit function:

[tex]P(43.5) = -43.5^2 + 87(43.5) - 300 = 1892.25[/tex]

Therefore, the maximum profit is $1,892,250.

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Answer:

option a

Step-by-step explanation:

it is the formula for varience.

The number of people who like the plan is 1,590 people out of the 3,000 surveyed.

To determine how many people liked the plan, we'll need to use the percentage given and apply it to the total number of people surveyed.

Percentage is a way of expressing a proportion or a fraction as a whole number out of 100. In this case, the percentage we're working with is 53%, which means 53 out of every 100 people surveyed liked the plan. To find the number of people who liked the plan, we can multiply the total number of people surveyed (3,000) by the percentage who liked the plan (53%).

To do this calculation, first convert the percentage to a decimal by dividing 53 by 100, which gives us 0.53. Next, multiply 3,000 by 0.53:

3,000 * 0.53 = 1,590

So, 1,590 people out of the 3,000 surveyed liked the plan for the new park.

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The mass of the thin bar with the given density function is 7/3.

To find the mass of the thin bar with the given density function p(x) = 2 + x^2 for 0 ≤ x ≤ 1:

You need to integrate the density function over the given interval.

Here are the steps to do that:

STEP 1: Set up the integral for mass:

Mass = ∫(density function) dx from x = 0 to x = 1
STEP 2:Plug in the given density function:

Mass = ∫(2 + x^2) dx from x = 0 to x = 1
STEP 3:Integrate the function with respect to x:

Mass = [2x + (x^3)/3] evaluated from x = 0 to x = 1
STEP 4: Evaluate the integral at the limits:

Mass = (2(1) + (1^3)/3) - (2(0) + (0^3)/3)
STEP 5: Simplify the expression:

Mass = (2 + 1/3) - 0
STEP 6: Calculate the final mass:

Mass = 2 + 1/3 = 7/3

So, the mass of the thin bar with the given density function is 7/3.\

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(a) Given: f(x) = x^2 + 4x + 3.

The partial fraction decomposition of f(x) is:

f(x) = (x+1)(x+3)

Now, we need to find the integral of this function from 1 to infinity:

∫[1,∞] (x+1)(x+3) dx

Since the integral converges, we can conclude that the series also converges.

(b) This series is not geometric, so we don't know what it converges to. However, we can decompose the given series as the difference of two sums:

Σ(n^2 + 4n + 3) = Σ(n^2) - Σ(4n)

(c) We can use index shifts to make these sums look similar enough to rewrite the expression without Σ:

Σ(n^2) - Σ(4n) = Σ(n^2 - 4n)

(d) To find the limit as B approaches 0, we can evaluate the limit of the expression n^2 + 4n + 3:

lim(B→0) (n^2 + 4n + 3) = n^2 + 4n + 3

So, the limit of the series is n^2 + 4n + 3.

Answer:

The answer is -84,484

Step-by-step explanation:

using Bodmas

multiplication first

5774+252-(2586×35)

5774+252-90510

6026-90510

-84,484

Answer: 1 cups of nuts : 1 1/2 cups of raisins

Step-by-step explanation:

S-au adus initial 68 de ghivece de flori, iar în fiecare zi s-au vândut, respectiv, 34, 17 și 17 ghivece.

Initial, la florărie s-au adus x ghivece de flori. În prima zi s-au vândut 1/2 * x ghivece. A doua zi, din numărul rămas s-au vândut 1/4 * (x - 1/2 * x) ghivece, adică 1/4 * 1/2 * x.

În plus față de acestea, s-au vândut încă 7 ghivece, deci în total în a doua zi s-au vândut 1/4 * 1/2 * x + 7 ghivece. În a treia zi s-au vândut restul de 20 de ghivece, deci numărul rămas la finalul celei de-a doua zile este x - 1/2 * x - 1/4 * 1/2 * x - 7. Trebuie să fie egal cu 20, deci avem ecuația x - 1/2 * x - 1/4 * 1/2 * x - 7 = 20.

Rezolvând această ecuație, obținem x = 128. Prin urmare, în prima zi s-au vândut 1/2 * 128 = 64 ghivece, în a doua zi s-au vândut 1/4 * 1/2 * 128 + 7 = 15 ghivece, iar în a treia zi s-au vândut restul, adică 20 ghivece.

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Answer:

Please look at the picture provided to see the correct triangle because each figure has no a,b, or c. Thanks

There is a probability that 105 of the 350 students would select May 9 for the dance.

To solve the above we need to find the proportion of students in the sample who voted for each date and this can be done by:

May 2:  18/70

  = 0.257

May 9: 39/70

  = 0.557

May 16: 13/70

   = 0.186

if the whole population of 350 students voted, then

May 2: 0.257 x 350

   =  90

May 9: 0.557 x 350

 = 195

May 16: 0.186 x 350

  = 65

From the above calculation, we can see that if 105 students out of 350) were to choose a date, the most likely date that they will select is  May 9, since it is the one that has the highest proportion of votes in the sample.

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Answer:

f(0) = 1

g(-2) = 3

f(-7)= und

g(4) x f(3) = -2 x 0 = 0

g(-4) = 2

g(x) = 0 --> x = 6, 0.5

f(x) = -1 --> x = -3, 5

f(g(3)) = f(-3) = -1

g(f(-2) = g(0) = -3

f(g(1)) = f(-3) = -1

f(g(5)) = f(-1) = 1

g(f(-4)) = g(-2) = 2

g(g(-6)) = g(4) = -2

g(f(0)) = g(1) = -3

g(f(-6)) = und

Step-by-step explanation:

In order to find the first group, such as f(0), you want to look at the f graph and find 0 on the x-axis. Wherever the y coordinate is will be the correct answer.

To find one such as f(g(3)), you want to dissect it like it is 2 problems. First, we want to find g(3) which is -3. Then we will find -3 on the f graph and find the answer with that y-coordinate.

673 children, 11 students, and 336 adults attended the movie.

How many children attended the movie?

How many students attended the movie?

How many adults attended the movie?

How to calculate the total ticket sales?

How to use equations to solve a word problem?

How to check if the obtained solution is valid?

Let's begin by defining some variables:

Let C be the number of children attending the movie.

Let S be the number of students attending the movie.

Let A be the number of adults attending the movie.

We know that the theater has a seating capacity of 323, so we can write an equation that relates the number of people attending the movie to the seating capacity:

C + S + A = 323

We also know that the theater charges $5.00 for children, $7.00 for students, and $12.00 for adults, and that there are half as many adults as there are children. Using this information, we can write another equation that relates the total ticket sales to the number of people in each category:

5C + 7S + 12A = 2348

We can use the fact that there are half as many adults as children to express A in terms of C:

A = 0.5C

Substituting this into the first equation, we get:

C + S + 0.5C = 323

Simplifying, we get:

1.5C + S = 323

Now we have two equations with two unknowns (C and S), which we can solve to find the values of these variables:

1.5C + S = 323 (equation 1)

5C + 7S = 2348 (equation 2)

Multiplying equation 1 by 5 and subtracting it from equation 2, we can eliminate S and solve for C:

5(1.5C + S) - 7S = 7.5C + 5S - 7S = 2348 - 5(323) = 1683

2.5C = 1683

C = 673.2

Since C must be a whole number, we can round down to the nearest integer:

C = 673

Now we can use this value of C to find S:

1.5C + S = 323

1.5(673) + S = 323

S = 323 - 1010.5

S = 10.5

Again, since S must be a whole number, we round up to the nearest integer:

S = 11

Finally, we can use the equation A = 0.5C to find A:

A = 0.5C = 0.5(673) = 336.5

Rounding down to the nearest integer, we get:

A = 336

Therefore, the number of children, students, and adults who attended the movie are:

673 children, 11 students, and 336 adults.

Any function of the form [tex]y = \sqrt[3]{x + a}[/tex] is a translation left a units of the graph of [tex]g(x) = \sqrt[3]{x}[/tex], which has one x-intercept.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The function in this problem has a single x-intercept, hence a translation left only moves the function laterally, meaning that it would also have only one x-intercept.

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The 90% confidence interval that estimates the average age of cars sold for the first time is (2.56, 10.30).

To calculate the confidence interval, we can use the formula:

CI =[tex]\bar{X}[/tex]  ± tα/2 * (s/√n)

where [tex]\bar{X}[/tex] is the sample mean, s is the sample standard deviation, n is the sample size, tα/2 is the critical value from the t-distribution table with (n-1) degrees of freedom and a confidence level of 90%.

Using the given data, we find that the sample mean is 6.43 years and the sample standard deviation is 2.69 years. With a sample size of 7, the critical value from the t-distribution table is 1.895.

Plugging in these values, we get:

CI = 6.43 ± 1.895 * (2.69/√7)

Simplifying this expression gives us the confidence interval (2.56, 10.30). Therefore, we can say with 90% confidence that the average age of cars sold for the first time is between 2.56 and 10.30 years.

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|4x +9|=|6-5x|

4x+5x=6-9

9x=-3

x=-3/9

x=-1/3

Answer:

x = 15

x = - ⅓

Step-by-step explanation:

|4x+9| = |6-5x|

4x+9=6-5x or 4x+9=-6+5x

Lets do the first one first

4x+9=6-5x

9x=-3

x= -3/9 = -1/3

4x+9=-6+5x

15=x

So the two solutions are x = 15 and x = -⅓

The required value of te angle ∠3 = 100 degree.

A circle is a shape that is made up of all of the points in a plane that are at a certain distance from the center. Alternatively, it is the plane-moving curve traced by a point at a constant distance from a given point.

From the figure of the circle, we can identify that AD and CD are two diameter of the circle.

and ∠COA = ∠3 is inscribed angle made by up by 2 radian of the circle

So, arcCA = ∠3 = 100 degree

Thus, required value of te angle ∠3 = 100 degree.

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The quadratic function for the graph and the duration the ball is in the air are;

Function; f(t) = -16·(t - h)² + k

Duration the ball is in the air is about 3.02 seconds

A quadratic function is a function that can be expressed in the form; f(x) = a·x² + b·x + c, where a ≠ 0, and a, b, and c are numbers.

The height at which the ball Alex releases the ball = 4 feet above the ground

The time it takes the ball to reach maximum height = 1.5 seconds

The required form of the function to be obtained based on the graph is f(t) = a·(t - h)² + k

f(t) = The height of the ball at time t

The required form of the function is the vertex form of a quadratic equation, where;

(h, k) = The coordinates of the vertex = (1.5, 40)

The points on the graph are; (0, 4), (3, 3)

Therefore; f(0) = a·(0 - 1.5)² + 40 = 4

a·(0 - 1.5)²  = 4 - 40 = -36

a = -36/(1.5²) = -16

The equation is; f(t) = -16·(t - 1.5)² + 40

The time the ball is in the air can be obtained from the function f(t) = -16·(t - 1.5)² + 40 as follows;

f(t) = -16·(t - 1.5)² + 40 = 3

-16·(t - 1.5)² = 3 - 40 = -37

(t - 1.5)² = -37/(-16)

(t - 1.5) = (√(37))/4

t = (√(37))/4 + 1.5 ≈ 3.02

The time the ball is in the air about 3.02 seconds

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Answer:

[tex]40 + 37 + 30 + 40 + 39 + 41 + 38 + n = 265 + n[/tex]

1)

[tex] \frac{265 + n}{8} = 43[/tex]

[tex]265 + n = 344[/tex]

[tex]n = 79[/tex]

2)

[tex] \frac{265 + n}{8} = 40[/tex]

[tex]265 + n = 320[/tex]

[tex]n = 55[/tex]

3)

[tex] \frac{265 + n}{8} = 38[/tex]

[tex]265 + n = 304[/tex]

[tex]n = 39[/tex]

The interest that has to be paid for the car is $ 4534.2.

Compound interest is a type of interest calculation in which the interest earned is added to the principal amount.

The principal is the sum that is left to be paid after the down payment hence;

Principal = $24, 599 - $5, 000

= $19599

A = P(1 + r/n)^nt

A = amount

P =principal

r = rate

n = Number of times compounded

t = time

Then we have that;

A = 19599( 1 + 0.0425)^5

A = $24133.2

Then ;

Interest = Amount - Principal

Interest = $24133.2 -  $19599

=$ 4534.2

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a) Approximately 79.38% of the phones have a battery life of at least 4 hours and 45 minutes.

b) Approximately 34.13% of the phones have a battery life between 4.5 hours and 5.25 hours.

c) Approximately 50% of the phones have a battery life less than 5 hours or greater than 5.5 hours.

a) What percentage battery life of 4 hours and 45 minutes?

a) For phones with a mean battery life of 5 hours and a standard deviation of 15 minutes, we can calculate the percentage of phones with a battery life of at least 4 hours and 45 minutes. By converting 4 hours and 45 minutes to minutes (4*60 + 45 = 285 minutes) and using the z-score formula, we find that the z-score is (285 - 300) / 15 = -1. Hence, the percentage is approximately 1 - 0.8359 = 0.1641, which is about 16.41%.

b) What percentage battery life between 4.5 hours and 5.25 hours?

b) To determine the percentage of phones with a battery life between 4.5 hours and 5.25 hours, we need to calculate the z-scores for both values. Converting the hours to minutes, we have 4.5 hours = 270 minutes and 5.25 hours = 315 minutes. The z-scores are (270 - 300) / 15 = -2 and (315 - 300) / 15 = 1. By referring to the standard normal distribution table, we find that the area between -2 and 1 is approximately 0.6141. Thus, the percentage is 0.6141 * 100 = 61.41%.

c) For the percentage of phones with a battery life less than 5 hours or greater than 5.5 hours, we need to calculate the z-score for both cases. The z-score for 5 hours is (300 - 300) / 15 = 0, and the z-score for 5.5 hours is (330 - 300) / 15 = 2. By referring to the standard normal distribution table, we find that the area to the left of 0 is 0.5 and the area to the right of 2 is 1 - 0.9772 = 0.0228. Adding these percentages, we get 0.5 + 0.0228 = 0.5228, which is approximately 52.28%.

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For each expression, the value is calculated by following the order of operations, i.e. first solving any multiplication or division, then addition or subtraction. The resulting values are: a + b - c = -3, a - b + c = -1, a + 2b - 2c = -5, (7b)/(b+c) = 3, ac + 2b - 2c = -10, and c(b-a) = 20.

To calculate a + b - c, we substitute a = -2, b = 3, and c = 4. So,

a + b - c = -2 + 3 - 4 = -3

To calculate a - b + c, we substitute a = -2, b = 3, and c = 4. So,

a - b + c = -2 - 3 + 4 = -1

To calculate a + 2b - 2c, we substitute a = -2, b = 3, and c = 4. So,

a + 2b - 2c = -2 + 2(3) - 2(4) = -5

To calculate (7b) / (b + c), we substitute b = 3 and c = 4. So,

(7b) / (b + c) = (7(3)) / (3 + 4) = 21 / 7 = 3

To calculate ac + 2b - 2c, we substitute a = -2, b = 3, and c = 4. So,

ac + 2b - 2c = (-2)(4) + 2(3) - 2(4) = -10

To calculate c(b - a), we substitute a = -2, b = 3, and c = 4. So,

c(b - a) = 4(3 - (-2)) = 4(5) = 20

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the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.

To find the marginal profit when 4 units are produced and sold, we first need to find the derivative of the profit function P(x) with respect to x. The given profit function is:

P(x) = (x^2) / (2x + 1)

Now, let's find its derivative, which represents the marginal profit function:

dP/dx = (d/dx(x^2))/(2x + 1) - (x^2)(d/dx(2x + 1))/((2x + 1)^2)

First, find the derivative of x^2 and 2x + 1:

d/dx(x^2) = 2x
d/dx(2x + 1) = 2

Now, substitute these values into the marginal profit function:

dP/dx = (2x)/(2x + 1) - (x^2)(2)/((2x + 1)^2)

Next, we'll find the marginal profit when 4 units are produced and sold. So, let's substitute x = 4 into the marginal profit function:

dP/dx(4) = (2 * 4)/(2 * 4 + 1) - (4^2)(2)/((2 * 4 + 1)^2)

dP/dx(4) = (8)/(9) - (32)(2)/(81)

dP/dx(4) = (8 - 64)/(81)

dP/dx(4) = -56/81

Since the profit is in hundreds of dollars, we need to multiply the marginal profit by 100 to get the value in dollars. Then, round to the nearest cent:

Marginal profit in dollars = (-56/81) * 100 ≈ -$69.14

So, the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.

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Jane has 12 coins and Jim has 29 coins.

We know that this is a word problem and the first thing that we have to do is to form the equation from the problem that have been given to us here. This is what we shall now proceed to do below.

Let the number of coins that Jane has be x

Number of coins that Jim has = 5 + 2x

Total number of coins = 41

Thus we have that;

x + 5 + 2x = 41

3x + 5 = 41

3x = 41 - 5

3x = 36

x = 12

This implies that Jane has 12 coins and Jim has 5 + 2(12) = 29 coins

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The expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27

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

(3^3*3^-4)^-3

Evaluating the expression in the brackets using the law of indices, we have

(3^3*3^-4)^-3 = (3^-1)^-3

Next, we open the brackets

This gives

(3^3*3^-4)^-3 = 3^(-1 *-3)

When evaluated, we have

(3^3*3^-4)^-3 = 3^3

Evaluate the exponents

This gives

(3^3*3^-4)^-3 = 27

Hence, the expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27

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The decision rule for rejecting the null hypothesis in this scenario is to reject the null hypothesis if the sample mean falls below a critical value determined by the level of significance and the population parameters.

To determine the decision rule for rejecting the null hypothesis, we need to conduct a hypothesis test. In this case, the null hypothesis (H0) is that the bag filling machine works correctly at the 449 gram setting. The alternative hypothesis (Ha) is that the machine is underfilling the bags.

Since the sample size is 24 and the population variance is unknown, we can use the t-distribution for the hypothesis test. With a level of significance of 0.1 (or 10%), the critical t-value can be obtained from the t-distribution table.

Using the sample mean of 445 grams, the sample variance of 196, and the sample size of 24, we can calculate the t-value. The decision rule is to reject the null hypothesis if the calculated t-value is less than the critical t-value or greater than the negative of the critical t-value.

To obtain the specific critical t-value, we need the degrees of freedom, which is (sample size - 1). In this case, it is 24 - 1 = 23. Consulting the t-distribution table or using statistical software, we can find the critical t-value corresponding to a 10% significance level and 23 degrees of freedom.

Finally, we compare the calculated t-value to the critical t-value to determine whether to reject the null hypothesis or not.

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There are 109 terms in given series.

Here, we have,

According to the statement

we have given that the sum of series is AP series and

This is 19 20.5 22 23.5 ... 181

And the sum of series is 10,900

Now, we have to find the number of terms in the series.

Then we use the summation formula which is

S = n/2 (a + L)

Substitute the all given values in it like

L = 181

A = 19 and S= 10,900

then

10,900= n/2(19+181)

10,900= n/2(200)

After solve the equation for n

10,900= 100n

n = 10,900 / 100

n = 109

There are 109 terms in given series.

So, There are 109 terms in given series.

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The total cost of repayment will be $7,737.00, and Marvin will pay $937.00 in interest over the 5-year period.

To find the total cost of repayment, we need to calculate the total amount that Marvin will pay over the course of 5 years. Since he is making monthly payments, we need to first find the total number of payments he will make:

Total number of payments = 5 years x 12 months/year = 60 payments

The total amount Marvin will pay is then:

Total amount = 60 payments x $128.95/payment = $7,737.00

To find the total interest Marvin will pay, we need to subtract the original amount of the loan from the total amount he will pay:

Total interest = Total amount - Loan amount

Total interest = $7,737.00 - $6,800.00

Total interest = $937.00

Therefore, the total cost of repayment will be $7,737.00, and Marvin will pay $937.00 in interest over the 5-year period.

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