Calculating the volume of a prism - GMAT Quantitative

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Question

What is the volume of a cube whose diagonal measures 10 inches?

Answer

By an extension of the Pythagorean Theorem, if is the length of an edge of the cube and is its diagonal length,

$d^{2}= s^{2}+s^{2}+s^{2} = 3s^{2}$

$\sqrt{3s^{2} }= \sqrt{d^{2}}$

$s\sqrt{3 }= d$

$s= \frac{d}{\sqrt{3 }}$

The volume is therefore

$V = s^{3} = \left (\frac{d}{\sqrt{3}} \right )^{3} = \left (\frac{10}{\sqrt{3}} \right )^{3}\approx 192.5$

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Question

A cube of iron has a mass of 4 kg. What is the mass of a rectangular prism of iron that is the same height but has a width and length that are twice as long as the cube?

Answer

Let the length of the sides of the cube equal 1. The volume of the cube is then $\dpi{100} 1\times 1\times 1=1$ . Therefore, the volume of the prism is $\dpi{100} 2\times 2\times 1=4$ . Therefore, the mass of the prism must be 4 times greater than the mass of the cube.

$\small (4)(4)=\ 16$

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Question

The sum of the length, the width, and the height of a rectangular prism is one yard. The length of the prism is eleven inches greater than its width, and the width is twice its height. What is the volume of the prism?

Answer

Let be the height of the prism. Then the width is , and the length is . Since the sum of the three dimensions is one yard, or 36 inches, we solve for in this equation:

$5h \div 5 = 25 \div 5$

The height is 5 inches; the width is twice this, or 10 inches; the length is eleven inches greater than the width, or 21 inches. The volume is the product of the three dimensions:

$V = lwh = 5 \cdot10\cdot21 = 1,050$ cubic inches.

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Question

The length, width, and height of a rectangular prism, in inches, are three different prime numbers. All three dimensions are between one yard and four feet. What is the volume of the prism?

Answer

One yard is equal to 36 inches; four feet are equal to 48 inches. There are four prime numbers between 36 and 48 - 37, 41, 43, and 47. Since we are only given that the dimensions are three different prime numbers between 36 and 48, we have no way of knowing which three they are.

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Question

A company makes cubic metal blocks whose edges measure 5 inches. The blocks are supposed to be packed in crates 40 inches long, 30 inches wide, and 32 inches high. How many blocks can be packed in one of these crates?

Answer

Each layer of blocks will be $40 \div 5 = 8$ blocks long and $30 \div 5 = 6$ blocks wide, making a layer of $8 \times 6 = 48$ blocks. The height is 32 feet, which divided by 5 is

$32 \div 5 = 6\textrm{ R }2$

so there will be 6 layers of blocks.

Therefore, $48 \times 6 = 288$ blocks can be packed.

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Question

A given rectangular prism has a length of , a width of , and a height of . What is the volume of the prism?

Answer

The volume of a given rectangular prism is the product of its length , width , and height , or . Plugging in the values provided:

$V=315cm^{3}$

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Question

A given cylinder has circular bases of radius and height . What is the volume of the cylinder?

Answer

The volume of a given cylinder is the product of its base area and height , or . Since the circular base area $A=\pi r^{2}$ , we can substitute and plug in the values provided:

$V=(\pi r^{2})h$

$V=(\pi (6)^{2})(12)$

$V=432\pi$

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Question

A given rectangular prism has a length of , a width of , and a height of . What is its volume?

Answer

The volume of a given rectangular prism is the product of its length , width , and height , or . Plugging in the values provided:

$V=1000cm^{3}$

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Question

Ron is making a wooden chest. The chest needs to measure 45 inches by 72 inches by 36 inches. What volume will the chest be able to hold in cubic feet?

Answer

Because we are asked for an answer in cubic feet, convert all measurments to feet right away

45 inches: 3.5 feet

72 inches: 6 feet

36 inches: 3 feet

Volume of a rectangular prism is given by:

V=lwh

So

V=363.5=67.5 ft^3

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Question

What is the volume of a cube whose diagonal measures 10 inches?

Answer

By an extension of the Pythagorean Theorem, if is the length of an edge of the cube and is its diagonal length,

$d^{2}= s^{2}+s^{2}+s^{2} = 3s^{2}$

$\sqrt{3s^{2} }= \sqrt{d^{2}}$

$s\sqrt{3 }= d$

$s= \frac{d}{\sqrt{3 }}$

The volume is therefore

$V = s^{3} = \left (\frac{d}{\sqrt{3}} \right )^{3} = \left (\frac{10}{\sqrt{3}} \right )^{3}\approx 192.5$

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Question

A cube of iron has a mass of 4 kg. What is the mass of a rectangular prism of iron that is the same height but has a width and length that are twice as long as the cube?

Answer

Let the length of the sides of the cube equal 1. The volume of the cube is then $\dpi{100} 1\times 1\times 1=1$ . Therefore, the volume of the prism is $\dpi{100} 2\times 2\times 1=4$ . Therefore, the mass of the prism must be 4 times greater than the mass of the cube.

$\small (4)(4)=\ 16$

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Question

The sum of the length, the width, and the height of a rectangular prism is one yard. The length of the prism is eleven inches greater than its width, and the width is twice its height. What is the volume of the prism?

Answer

Let be the height of the prism. Then the width is , and the length is . Since the sum of the three dimensions is one yard, or 36 inches, we solve for in this equation:

$5h \div 5 = 25 \div 5$

The height is 5 inches; the width is twice this, or 10 inches; the length is eleven inches greater than the width, or 21 inches. The volume is the product of the three dimensions:

$V = lwh = 5 \cdot10\cdot21 = 1,050$ cubic inches.

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Question

The length, width, and height of a rectangular prism, in inches, are three different prime numbers. All three dimensions are between one yard and four feet. What is the volume of the prism?

Answer

One yard is equal to 36 inches; four feet are equal to 48 inches. There are four prime numbers between 36 and 48 - 37, 41, 43, and 47. Since we are only given that the dimensions are three different prime numbers between 36 and 48, we have no way of knowing which three they are.

Compare your answer with the correct one above

Question

A company makes cubic metal blocks whose edges measure 5 inches. The blocks are supposed to be packed in crates 40 inches long, 30 inches wide, and 32 inches high. How many blocks can be packed in one of these crates?

Answer

Each layer of blocks will be $40 \div 5 = 8$ blocks long and $30 \div 5 = 6$ blocks wide, making a layer of $8 \times 6 = 48$ blocks. The height is 32 feet, which divided by 5 is

$32 \div 5 = 6\textrm{ R }2$

so there will be 6 layers of blocks.

Therefore, $48 \times 6 = 288$ blocks can be packed.

Compare your answer with the correct one above

Question

A given rectangular prism has a length of , a width of , and a height of . What is the volume of the prism?

Answer

The volume of a given rectangular prism is the product of its length , width , and height , or . Plugging in the values provided:

$V=315cm^{3}$

Compare your answer with the correct one above

Question

A given cylinder has circular bases of radius and height . What is the volume of the cylinder?

Answer

The volume of a given cylinder is the product of its base area and height , or . Since the circular base area $A=\pi r^{2}$ , we can substitute and plug in the values provided:

$V=(\pi r^{2})h$

$V=(\pi (6)^{2})(12)$

$V=432\pi$

Compare your answer with the correct one above

Question

A given rectangular prism has a length of , a width of , and a height of . What is its volume?

Answer

The volume of a given rectangular prism is the product of its length , width , and height , or . Plugging in the values provided:

$V=1000cm^{3}$

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Question

Ron is making a wooden chest. The chest needs to measure 45 inches by 72 inches by 36 inches. What volume will the chest be able to hold in cubic feet?

Answer

Because we are asked for an answer in cubic feet, convert all measurments to feet right away

45 inches: 3.5 feet

72 inches: 6 feet

36 inches: 3 feet

Volume of a rectangular prism is given by:

V=lwh

So

V=363.5=67.5 ft^3

Compare your answer with the correct one above

Question

What is the volume of a cube whose diagonal measures 10 inches?

Answer

By an extension of the Pythagorean Theorem, if is the length of an edge of the cube and is its diagonal length,

$d^{2}= s^{2}+s^{2}+s^{2} = 3s^{2}$

$\sqrt{3s^{2} }= \sqrt{d^{2}}$

$s\sqrt{3 }= d$

$s= \frac{d}{\sqrt{3 }}$

The volume is therefore

$V = s^{3} = \left (\frac{d}{\sqrt{3}} \right )^{3} = \left (\frac{10}{\sqrt{3}} \right )^{3}\approx 192.5$

Compare your answer with the correct one above

Question

A cube of iron has a mass of 4 kg. What is the mass of a rectangular prism of iron that is the same height but has a width and length that are twice as long as the cube?

Answer

Let the length of the sides of the cube equal 1. The volume of the cube is then $\dpi{100} 1\times 1\times 1=1$ . Therefore, the volume of the prism is $\dpi{100} 2\times 2\times 1=4$ . Therefore, the mass of the prism must be 4 times greater than the mass of the cube.

$\small (4)(4)=\ 16$

Compare your answer with the correct one above

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