Volume of Other Solids - GED Math

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Question

The above depicts a rectangular swimming pool for an apartment.

On the right edge, the pool is three feet deep; on the left edge, it is eight feet deep. Going from the right to the left, its depth increases uniformly. In cubic feet, how much water does the pool hold?

Answer

The pool can be looked at as a trapezoidal prism with "height" 35 feet and its bases the following shape (depth exaggerated):

The area of this trapezoidal base, which has height 50 feet and bases 3 feet and 8 feet, is

$B = \frac{1}{2} (3+8) \cdot 50 = 275$ square feet;

The volume of the pool is the base multiplied by the "height", or

$275 \cdot 35 = 9,625$ cubic feet, the capacity of the pool.

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Question

Refer to the right circular cone in the above diagram. What is its volume, to the nearest whole cubic centimeter?

Answer

The volume of a right circular cone of height and base with radius is

$V = \frac{1}{3} \pi r^{2}h$ .

The radius is 50. To find the height, we need to use the Pythagorean Theorem with the radius 50 as one leg and the slant height 130 as the hypotenuse of a right triangle, and the height as the other leg:

$h^{2} + 50^{2} = 130 ^{2}$

$h^{2} + 2,500= 16,900$

$h^{2} + 2,500- 2,500= 16,9 00 - 2,500$

$h^{2} = 14,4 00$

$h = \sqrt{14,400} = 120$

Substitute 120 for and 50 for in the volume formula:

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \times 3.14 \times 50^{2} \times 120$

$V = \frac{1}{3} \times 3.14 \times 2,500 \times 120 \approx 314,159$ cubic centimeters.

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Question

Find the volume of a cube with a height of 12cm.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 12cm. Because it is a cube, all sides (lengths, widths, heights) are equal. Therefore, the length and the width are also 12cm. So, we can substitute. We get

$V = 12\text{cm} \cdot12\text{cm} \cdot12\text{cm}$

$V = 144\text{cm}^2 \cdot 12\text{cm}$

$V = 1728\text{cm}^3$

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Question

A cone has a diameter of 10in and a height of 6in. Find the volume.

Answer

To find the volume of a cone, we will use the following formula:

$V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 10in. We also know the diameter is two times the radius. Therefore, the radius is 5in.

We know the height of the cone is 6in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (5\text{in})^2 \cdot \frac{6\text{in}}{3}$

$V = \pi \cdot 25\text{in}^2 \cdot 2\text{in}$

$V = \pi \cdot 50\text{in}^3$

$V = 50\pi \text{ in}^3$

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Question

Find the volume of a cube with a height of 11in.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 11in. Because it is a cube, all sides are equal. Therefore, the width and the length are also 11in. So, we can substitute. We get

$V = 11\text{in} \cdot11\text{in} \cdot11\text{in}$

$V = 121\text{in}^2 \cdot 11\text{in}$

$V = 1331\text{in}^3$

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Question

Find the volume of a cube with a height of 10cm.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 10cm. Because it is a cube, all sides (lengths, widths, heights) are equal. Therefore, the length and the width are also 10cm. So, we can substitute. We get

$V = 10\text{cm} \cdot 10\text{cm} \cdot 10\text{cm}$

$V = 100\text{cm}^2 \cdot 10\text{cm}$

$V = 1000\text{cm}^3$

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Question

A cone has a diameter of 8in and a height of 9in. Find the volume.

Answer

To find the volume of a cone, we will use the following formula:

$V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 8in. We also know the diameter is two times the radius. Therefore, the radius is 4in.

We know the height of the cone is 9in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (4\text{in})^2 \cdot \frac{9\text{in}}{3}$

$V = \pi \cdot 16\text{in}^2 \cdot 3\text{in}$

$V = \pi \cdot 48\text{in}^3$

$V = 48\pi \text{ in}^3$

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Question

Find the volume of a cube with a height of 7in.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 7in. Because it is a cube, all sides are equal. Therefore, the width and the length are also 7in. So, we can substitute. We get

$V = 7\text{in} \cdot 7\text{in} \cdot 7\text{in}$

$V = 49\text{in}^2 \cdot 7\text{in}$

$V = 343\text{in}^3$

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Question

Let $\pi = 3.14$ .

A cone has a height of 5in and a diameter of 12in. Find the volume.

Answer

To find the volume of a cone, we will use the following formula:

$V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

We know $\pi=3.14$ .

We know the diameter of the cone is 12in. We know the diameter is two times the radius. So, the radius is 6in.

We know the height is 5in.

Now, we can substitute. We get

$V = 3.14 \cdot (6\text{in})^2 \cdot \frac{5\text{in}}{3}$

$V = 3.14 \cdot 36\text{in}^2 \cdot \frac{5\text{in}}{3}$

Now, we can simplify to make things easier. The 36 and the 3 can both be divided by 3. So, we get

$V = 3.14 \cdot 12\text{in}^2 \cdot \frac{5\text{in}}{1}$

$V = 3.14 \cdot 12\text{in}^2 \cdot 5\text{in}$

$V = 3.14 \cdot 60\text{in}^3$

$V = 188.4\text{in}^3$

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Question

If a cube has a width of 7in, find the volume.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

We know the width of the cube is 7in. Because it is a cube, all sides/lengths are equal. Therefore, the length and the height are also 7in.

Now, we can substitute. We get

$V = 7\text{in} \cdot 7\text{in} \cdot 7\text{in}$

$V = 7\text{in} \cdot 49\text{in}^2$

$V = 343\text{in}^3$

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Question

If a cube has a height of 8cm, find the volume.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 8cm. Because it is a cube, all sides/lengths are the same. Therefore, the length and the width are also 8cm. So, we can substitute. We get

$V = 8\text{cm} \cdot 8\text{cm} \cdot 8\text{cm}$

$V = 64\text{cm}^2 \cdot 8\text{cm}$

$V = 512\text{cm}^3$

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Question

Let $\pi = 3.14$ .

Find the volume of a cone with the following measurements:

radius: 7cm

height: 9cm

Answer

To find the volume of a cone, we will use the following formula:

$V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know $\pi = 3.14$ . We know the radius of the cone is 7cm. We know the height of the cone is 9cm. So, we substitute. We get

$V = 3.14 \cdot (7\text{cm})^2 \cdot \frac{9\text{cm}}{3}$

$V = 3.14 \cdot 49\text{cm}^2 \cdot \frac{9\text{cm}}{3}$

$V = 3.14 \cdot 49\text{cm}^2 \cdot 3\text{cm}$

$V = 3.14 \cdot 147\text{cm}^3$

$V = 461.58\text{cm}^3$

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Question

Find the volume of a cube with a width of 11in.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the width of the cube is 11in. Because it is a cube, all widths/lengths/etc are the same. Therefore, the length and the height are also 11in. So, we substitute. We get

$V = 11\text{in} \cdot 11\text{in} \cdot 11\text{in}$

$V = 121\text{in}^2 \cdot 11\text{in}$

$V = 1331\text{in}^3$

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Question

Find the volume of a cone with a radius of 2 and a height of 10.

Answer

Write the formula for the volume of a cone.

$V =\frac{1}{3}\pi r^2 h$

Substitute the radius and height.

$V =\frac{1}{3}\pi (2)^2 (10) = \frac{40}{3}\pi$

The answer is: $\frac{40}{3}\pi$

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Question

Find the volume of a cone with a radius of $6\pi$ and a height of .

Answer

Write the formula for the volume of a cone.

$V = \frac{1}{3}\pi r^2h$

Substitute the radius and height into the formula.

$V = \frac{1}{3}\pi (6\pi)^2(2) = \frac{1}{3}\pi (6\pi)(6\pi)(2)$

The answer is: $24 \pi ^3$

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Question

What is the volume of a hemisphere with a radius of 2?

Answer

Recall that the volume of a full sphere is: $V= \frac{4}{3} \pi r^3$

A hemisphere would be half this volume.

$\frac{1}{2}(\frac{4}{3} \pi r^3) = \frac{2}{3} \pi r^3$

Substitute the radius.

$\frac{2}{3} \pi (2)^3 = \frac{2}{3} \pi (8) = \frac{16}{3}\pi$

The answer is: $\frac{16}{3}\pi$

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Question

A cube has a length of 9cm. Find the volume.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the length of the cube is 9cm. Because it is a cube, all sides/lengths are equal. Therefore, the width and height are also 9cm.

Knowing this, we can substitute into the formula. We get

$V = 9\text{cm} \cdot 9\text{cm} \cdot 9\text{cm}$

$V = 81\text{cm}^2 \cdot 9\text{cm}$

$V = 729\text{cm}^3$

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Question

Find the volume of a cube with a height of 6cm.

Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 6cm. Because it is a cube, all lengths/sides/etc are equal. Therefore, the length and the width are also 6cm.

So, we substitute. We get

$V = 6\text{cm} \cdot 6\text{cm} \cdot 6\text{cm}$

$V = 36\text{cm}^2 \cdot 6\text{cm}$

$V = 216\text{cm}^3$

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Question

Find the volume of a cone with a radius of 8in and a height of 6in.

Answer

To find the volume of a cone, we will use the following formula:

$V = \pi r^2 \frac{h}{3}$

where r is the radius and h is the height of the cone.

Now, we know the radius is 8in. We know the height is 6in. So, we can substitute. We get

$V = \pi \cdot (8\text{in})^2 \cdot \frac{6\text{in}}{3}$

$V = \pi \cdot 64\text{in}^2 \cdot 2\text{in}$

$V = \pi \cdot 128\text{in}^3$

$V = 128\pi \text{ in}^3$

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Question

An office uses cone-shaped paper cups for water in their water cooler. The cups have a radius of inches and a height of inches. If the water cooler can hold cubic inches of water, how many complete cups of water can the water cooler fill?

Answer

Start by finding the volume of a cup.

Recall how to find the volume of a cone:

$\text{Volume of Cone}=\frac{1}{3}\pi r^2 h$

Plug in the given radius and height to find the volume.

$\text{Volume}=\frac{1}{3}\pi(3)^2(4.25)=40.055$

Now divide the total volume of the water in the water cooler by the volume of one cup in order to find how many complete cups the water cooler can fill.

$\text{Number of cups filled}=\frac{1155}{40.055}=28.84$

Since the question asks for the number of complete cups that can be filled, we must round down to .

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