Prisms - GMAT Quantitative

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Question

If a rectangular prism has a length of , a width of , and a height of , what is the length of its diagonal?

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle formed by the height of the prism and the diagonal of its bottom face. First apply the Pythagorean Theorem to find the length of the diagonal of the bottom face, and then apply the Pythagorean Theorem again with this side and the height of the prism to find the length of its diagonal:

$C^2=52\rightarrow C=\sqrt{52}$

$3^2+(\sqrt{52})^2=D^2$

$D^2=61\rightarrow D=\sqrt{61}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle, where the other two sides are the height of the prism and the diagonal of either its top or bottom face. This means we can find the length of the prism's diagonal using the Pythagorean Theorem, but first we must apply this theorem to find the diagonal of the prism's top or bottom face, which forms the base of the right triangle whose hypotenuse is the diagonal of the entire prism. After finding the face diagonal, we apply the Pythagorean Theorem again to calculate the answer:

$c^2=89\rightarrow c=\sqrt{89}$

$3^2+(\sqrt{89})^2=d^2$

$d^2=98\rightarrow d=\sqrt{98}=\sqrt{2\cdot 49}=7\sqrt{2}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle whose other two sides are the height of the prism and the diagonal of either its top or bottom face. We start by finding the diagonal of either the prism's top or bottom face, as this is the base of the right triangle for which the diagonal of the prism is the hypotenuse:

$c^2=25\rightarrow c=5$

Now we apply the Pythagorean Theorem again, this time using the prism height and the face diagonal calculated above, and the hypotenuse we're left with is the same as the diagonal length of the rectangular prism:

$c^2=29\rightarrow c=\sqrt{29}$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{12^{2} +4^{2}}=c$

$\sqrt{144 +16}=c$

$\sqrt{160}=c$

$\sqrt{16\times 10}=c$

$4\sqrt{10}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{9^{2} +(\sqrt{160})^{2}}=d$

$\sqrt{81 +160}=d$

$\sqrt{241}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{8^{2} +10^{2}}=c$

$\sqrt{64 +100}=c$

$\sqrt{164}=c$

$\sqrt{41\times 4}=c$

$2\sqrt{41}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{6^{2} +(\sqrt{164})^{2}}=d$

$\sqrt{36 +164}=d$

$\sqrt{200}=d$

$\sqrt{100\times 2}=d$

$10\sqrt{2}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{7^{2} +4^{2}}=c$

$\sqrt{49 +16}=c$

$\sqrt{65}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{11^{2} +\sqrt{65}^{2}}=d$

$\sqrt{121+65}=d$

$\sqrt{186}=d$

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Question

Find the diagonal of a rectangular prism whose base is $3\times4$ and has a base of .

Answer

To solve, simply solve for the base diagonal which will become the side of the other triangle, whose hypotenuse is the diagonal we are looking for.

$c1^2=3^2+4^2\Rightarrow c=5$

$diagonal=\sqrt{5^2+12^2}=13$

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Question

A new fish tank at a theme park must hold 450,000 gallons of sea water. Its dimensions must be such that it is twice as long as it is wide, and half as high as it is wide. If one gallon of water occupies 0.1337 cubic feet, then give the surface area of the proposed tank to the nearest square foot.

You may assume that the tank has all four sides and a bottom, but is open at the top.

Answer

450,000 gallons of water occupy $\small \small 0.1337 \cdot 450,000 \approx 60,165$ cubic feet.

Let $\small h$ be the height of the tank. Then the width of the tank is , and its length is . Multiply the dimensions to get the volume:

$V = LWH = H \cdot 2H\cdot 4H = 8H^{3} = 60,165$

$H^{3} = 60,165 \div 8 \approx 7,521$

$H = \sqrt[3]{7,521} \approx 19.6$

$W = 2H = 2 \cdot19.6 = 39.2$

$L = 2W = 2\cdot 39.2 = 78.4$

Since the tank has four sides and a bottom, but not a top, its surface area is

$A = 2\cdot 78.4\cdot 19.6 + 2\cdot 39.2\cdot 19.6 + 78.4\cdot 39.2$

$A = 2\cdot 78.4\cdot 19.6 + 2\cdot 39.2\cdot 19.6 + 78.4\cdot 39.2 \approx 7,683$

The surface area of the tank is about 7,683 square feet.

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Question

A rectangular prism has a volume of . If the length of the prism is and its width is , what is its height?

Answer

The volume of a rectangular prism is equal to its length times its width times its height. We are given the volume, the length, and the width, so using the following formula we can solve for the height of the rectangular prism:

$H=\frac{105}{35}=3$

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Question

Jenny wants to make a cube out of sheet metal. What is the length of one side of the cube?

I) The cube will require square inches of material.

II) The cube will hold cubic inches.

Answer

The length of an edge of a cube can be used to find either volume or surface area, and vice versa.

I) Gives us the surface area thus, we are able to calculate the length of an edge using the formula,

$SA=6\cdot s^2$

$294=6s^2 \rightarrow 49=s^2 \rightarrow 7=s$ .

II) Gives us the volume thus, we are able to calculate the length of an edge using the formula,

$343=s^3 \rightarrow 7=s$

Either can be used to find the side length.

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

Answer

Using the formula for the volume of a rectangular prism, we can plug in the given values and solve for the width of the prism:

$W=\frac{70}{14}=5$

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Question

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

Answer

We are given the surface area and the length of two sides, so in order to calculate the length of the third side we need the formula for the surface area of a prism in terms of each side length:

Using the formula, we can simply plug in the given values and solve for the length of the rectangular prism:

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Question

The volume of the rectangular solid above is 120. The area of ABFE is 20 and the area of ABCD is 30. What is the area of BFGC?

Answer

We can set the lengths of three sides to be , , , respectively. The volume is 120 means that . Also we know the areas of two sides, so we can use to represent the area of 20 and to represent the area of 30. Now the question is to figure out . Then we can use

to solve .

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Question

What is the surface area of a rectangular prism that is 4 inches long, 6 inches wide, and 5 inches high?

Answer

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Question

A box with dimensions 8 inches, 10 inches, and 5 inches needs to be gift wrapped. Gift wrapping is priced at $0.10 per square inch of surface of a box. How much will it cost to wrap the gift?

Answer

Find the surface area of the box by summing the area of all six faces: two 8 by 10, two 8 by 5, two 10 by 5.

$2(80)+2(40)+2(50)=340\ in^{2}$

Since the price is $.10 for each square inch,

$Total\ cost=340(.10)=$34$

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Question

The sum of the length, the width, and the height of a rectangular prism is one meter. The length of the prism is sixteen centimeters greater than its width, which is three times its height. What is the surface area of this 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 meter, or 100 centimeters, we solve for in this equation:

$7h \div 7 = 84 \div 7$

The height of the prism is 12 cm; the width is three times this, or 36 cm; the length is sixteen centimeters greater than the width, which is 52 cm.

Set in the formula for the surface area of a rectangular prism:

$= 2 \cdot 52 \cdot 36 + 2 \cdot 36 \cdot 12 + 2 \cdot 52 \cdot 12$

square centimeters

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Question

A right prism has as its bases two equilateral triangles, each of whose sides has length 6. The height of the prism is three times the perimeter of a base. Give the surface area of the prism.

Answer

The area of each equilateral triangle base can be determined by setting in the formula

$B= \frac{s^{2}\sqrt{3}}{4} = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9\sqrt{3}$

The perimeter of each base is $P=6 \cdot 3 = 18$ , and the height of the prism is three times this, or $h = 18 \cdot 3 = 54$ . The lateral area of a prism is equal to the perimeter of a base multiplied by the height, so

$L =Ph= 18 \cdot 54 = 972$

Add this to the areas of two bases - the surface area is

$S= L+ B + B= 972 + 9\sqrt{3}+ 9\sqrt{3}= 972 + 18 \sqrt{3}$ .

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Question

A right prism has as its bases two right triangles, each of whose legs have lengths 12 and 16. The height of the prism is half the perimeter of a base. Give the surface area of the prism.

Answer

The area of a right triangle is equal to half the product of its legs, so each base has area

$\frac{1}{2} \cdot 12 \cdot 16 = 96$

The measure of the hypotenuse of each base is determined using the Pythagorean Theorem:

$c = \sqrt{12^{2}+16^{2}}= \sqrt{144+256} = \sqrt{400}= 20$

Therefore, the perimeter of each base is

,

and the height of the prism is half this, or $h = \frac{1}{2} \cdot 48 = 24$ .

The lateral area of the prism is the product of its height and the perimeter of a base; this is

$L= Ph= 24 \cdot 48 = 1,152$

The surface area is the sum of the lateral area and the two base areas, or

.

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Question

A right prism has as its bases two isosceles right triangles, each of whose hypotenuse has length 10. The height of the prism is the length of one leg of a base. Give the surface area of the prism.

Answer

By the 45-45-90 Theorem, dividing the length of the hypotenuse of an isosceles right triangle by $\sqrt{2}$ yields the length of one leg; therefore, the length of one leg of each base is

$\frac{10}{\sqrt{2}} = \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10 \sqrt{2}}{2} = 5\sqrt{2}$ .

The area of a right triangle is half the product of its legs, so the area of each base is

$B = \frac{1}{2} \cdot 5 \sqrt{2} \cdot 5 \sqrt{2} = \frac{1}{2} \cdot 25 \cdot 2 = 25$

The perimeter of each base is the sum of its sides, which here is

$P = 5 \sqrt{2}+ 5 \sqrt{2}+ 10 = 10 + 10 \sqrt{2}$

The height of the prism is the length of one leg of a base, which is $h = 5\sqrt{2}$ .

The lateral area of the prism is equal to the product of the height of the prism and the perimeter of a base, so

$= \left (10+10 \sqrt{2} \right ) \cdot 5 \sqrt{2}$

$= 10 \cdot 5 \sqrt{2} +10 \sqrt{2} \cdot 5 \sqrt{2}$

$= 50 \sqrt{2} +10\cdot 5 \cdot 2$

$= 100+ 50 \sqrt{2}$

The surface area is the sum of the lateral area and the areas of the bases:

$S=L+B+B= 100+ 50 \sqrt{2}+25+25 = 150+ 50 \sqrt{2}$

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