Prisms

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GMAT Quantitative › Prisms

Questions 1 - 10

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

$192.5; in ^{3}$

$100; in ^{3}$

$1,000; in ^{3}$

$333.3 ;in^{3}$

$353.6;in^{3}$

Explanation

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

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

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.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Explanation

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.

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?

Explanation

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.

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

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.

$\small 7,683 ; ft ^{2}$

$\small 10,756 ; ft ^{2}$

$\small 5,378 ; ft ^{2}$

$\small 9,220 ; ft ^{2}$

$\small 9,988 ; ft ^{2}$

Explanation

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.

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?

$5,856 \textrm{ cm}^{2}$

$22,464 \textrm{ cm}^{2}$

$25,872 \textrm{ cm}^{2}$

$6,104 \textrm{ cm}^{2}$

$7,672 \textrm{ cm}^{2}$

Explanation

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

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?

It is impossible to tell from the information given.

$71,299\textrm{ in}^{3}$

$65,231\textrm{ in}^{3}$

$82,861\textrm{ in}^{3}$

$74,777\textrm{ in}^{3}$

Explanation

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.

A right prism has as its bases two right triangles, each of which has a hypotenuse of length 20 and a leg of length 10. The height of the prism is equal to the length of the longer leg of a base. Give the surface area of the prism.

$300+400 \sqrt{3}$

$400+300 \sqrt{3}$

$500+300 \sqrt{3}$

Explanation

A right triangle with one leg half the length of the hypotenuse is a 30-60-90 triangle. Its other leg has measure $\sqrt{3}$ times the length of the first leg, which in the case of each base is $10 \sqrt{3}$ .

The area of each base is half the product of the legs, or

$B = \frac{1}{2} \cdot 10 \cdot 10 \sqrt{3} = 50 \sqrt{3}$ .

The perimeter of each base is

$P = 20+10+10 \sqrt {3} = 30 + 10 \sqrt {3}$

and the height of the prism is equal to the length of the longer leg, or

$h= 10 \sqrt{3}$ .

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

$=\left ( 30 + 10 \sqrt {3} \right ) \cdot 10 \sqrt {3}$

$= 30 \cdot 10 \sqrt {3}+ 10 \sqrt {3}\cdot 10 \sqrt {3}$

$= 30 \cdot 10 \sqrt {3}+ 10 \cdot 10\cdot \sqrt {3}\cdot \sqrt {3}$

$= 30 \cdot 10 \sqrt {3}+ 10 \cdot 10\cdot 3$

$= 300+ 30 0 \sqrt {3}$

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

$S=L+B+B= 300+300 \sqrt{3} + 50 \sqrt{3} + 50 \sqrt{3}= 300+400 \sqrt{3}$

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

$\sqrt{61}$

$2\sqrt{13}$

$\sqrt{71}$

$15\sqrt{2}$

Explanation

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:

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

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

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

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.

Explanation

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

A right prism has as its bases two isosceles right triangles, each of whose legs has length 16. The height of the prism is the length of the hypotenuse of a base. Give the surface area of the prism.

$768 + 512 \sqrt{2}$

$512+ 512 \sqrt{2}$

$640+ 512 \sqrt{2}$

$256+ 512 \sqrt{2}$

$384+ 512 \sqrt{2}$

Explanation

By the 45-45-90 Theorem, multiplying the length of a leg of an isosceles right triangle by $\sqrt{2}$ yields the length of its hypotenuse; therefore, the length of the hypotenuse of each base is $16 \sqrt{2}$ .