How to find the volume of a figure - HSPT Math

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Question

I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides. If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube?

Answer

Determine the volume of both cubes and then subtract the smaller from the larger. The large cube volume is 9” * 9” * 9” = 729 in3 and the small cube is 3” * 3” * 3” = 27 in3. The difference is 702 in3.

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Question

A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long?

Answer

A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.

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Question

The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?

Answer

Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:

V = π(d/2)2h = πd2h/4

Sub in h and V: 36p = πd2(4)/4 so 36p = πd2

Thus d = 6

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Question

The density of gold is $16\ g/cm^3$ and the density of glass is $2\ g/cm^3$ . You have a gold cube that is $x\ cm$ in length on each side. If you want to make a glass cube that is the same weight as the gold cube, how long must each side of the glass cube be?

Answer

Weight = Density * Volume

Volume of Gold Cube = side3= x3

Weight of Gold = 16 g/cm3 * x3

Weight of Glass = 3/cm3 * side3

Set the weight of the gold equal to the weight of the glass and solve for the side length:

16* x3 = 2 * side3

side3 = 16/2* x3 = 8 x3

Take the cube root of both sides:

side = 2x

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Question

Chemicals to clean a swimming pool cost \$0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

Answer

The volume of the pool can be determined by multiplying the length, width, and height together.

$V = 6 \times 8 \times 14 = 672 ft^{3}$

Each cubit foot costs 24 cents, so:

$672ft^3\times $0.24=$161.28\approx \$161$

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Question

What is the volume of a box with a length of 5, a height of 7, and a base of 16?

Answer

When searching for the volume of a box we are looking for the amount of the space enclosed by the box. To find this we must know the formula for the volume of a box which is

Using this formula we plug in the numbers for Base, Height, and Length to get

Multiply to arrive at the answer of .

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Question

The above depicts a rectangular swimming pool for an apartment. The pool is two meters deep everywhere. What is the volume of the pool in cubic meters?

Answer

The pool can be seen as a rectangular prism with dimensions 24 meters by 15 meters by 2 meters; its volume is the product of these dimensions, or

$24 \times 15 \times 2 = 720$ cubic meters.

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Question

One cubic meter is equal to one thousand liters.

The above depicts a rectangular swimming pool for an apartment. The pool is meters deep everywhere. How many liters of water does the pool hold?

Answer

The pool can be seen as a rectangular prism with dimensions meters by meters by meters; its volume in cubic meters is the product of these dimensions, which is

$24 \times 15 \times 2.5 = 900$ cubic meter.

One cubic meter is equal to one thousand liters, so multiply:

$900 \times 1,000 = 900,000$ liters of water.

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Question

A cheese seller has a 2 foot x 2 foot x 2 foot block of gouda and she wants to cut it into smaller gouda cubes that are 1.5 inches on a side. How many cubes can she cut?

Answer

First we need to determine how many of the small cubes of gouda would fit along one dimension of the large cheese block. One edge of the large block is 24 inches, so 16 smaller cubes $(24\div 1.5)$ would fit along the edge. Now we simply cube this one dimension to see how many cubes fit within the whole cube. $16^{3}=4096$ .

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Question

An aquarium is shaped like a perfect cube; the area of each glass face is 1.44 square meters. If it is filled to the recommended 90% capacity, then, to the nearest hundred liters, how much water will it contain?

Note: 1 cubic meter = 1,000 liters.

Answer

A perfect cube has square faces; if a face has area 1.44 square meters, then each side of each face measures the square root of this, or 1.2 meters. The volume of the tank is the cube of this, or

$1.2^{3} = 1.728$ cubic meters.

Its capacity in liters is $1.728 \times 1,000 = 1,728$ liters.

90% of this is

$1,728 \times 0.9 = 1,555.2$ liters.

This rounds to 1,600 liters, the correct response.

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Question

What is the volume of a cylinder with a diameter of 6, and a height of 5?

Answer

Write the formula to find the volume of a cylinder.

$V=\pi r^2 h$

The radius is half the diameter, which is 3.

Substitute all the known dimensions into the formula.

$V=\pi( 3^2) (5)=45\pi$

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Question

Find the volume of the cube if the face of square has a perimeter of $\textup{4 units}$ .

Answer

Find the side length of the square given the square perimeter. The perimeter of a square is:

The side of the cube has a length of one.

Write the formula to find the volume of a cube.

Substitute the side length.

$V=s^3= (1)^3=1\textup{ units}^{3}$

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=9\times\2\times3$

$\small v=54cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=7\times\2\times4$

$\small v=56cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=5\times\2\times3$

$\small v=30cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=9\times\2\times5$

$\small v=90cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=7\times\3\times4$

$\small v=84cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=3\times\1\times2$

$\small v=6cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=9\times\4\times5$

$\small v=180cm^3$

Remember, volume is always labeled as units to the third power.

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Question

What is the volume of the rectangular prism below?

Answer

The formula for volume of a rectangular prism is $\small v=l\times w\times h$

$\small v=6\times\1\times3$

$\small v=18cm^3$

Remember, volume is always labeled as units to the third power.

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