A rectangular solid has a surface area of 384 square centimeters. Which of the following could be the solid's volume?

Question

A rectangular solid has a surface area of 384 square centimeters. Which of the following could be the solid's volume?

 

• A) 500 cubic cm
• B) 600 cubic cm
• C) 700 cubic cm
• D) 800 cubic cm
• E) 900 cubic cm

Answer 1

For a rectangular solid, the maximum volume it can have, given that you know it's surface area, is for the solid to be a cube.

 

So let's suppose that this solid IS a cube and see what happens.

 

The surface area of a cube is 6s² where s is the length of an edge.  If 6s² = 384, then we can solve this to fund that s = 8.  In this case, the volume is 8³ = 512.

 

So 512 is the maximum volume that the solid can have.  Answers B, C, D, and E are volumes that are GREATER than this maximum, which is impossible.  The only answer that is less than or equal to the maximum is A, 500.

 

Answer 2

We know that a rectangular solid has three dimensions: length, width, and height.  This solid could actually be a cube, so the three dimensions are equal to each other. 

 

Let x = the side lengths of the solid

 

 

Surface area is the sum of the areas of all the rectangular faces of the solid.  Since a rectangular solid has 6 faces, we can say that

 

SA = 6x²

 

6x² = 384

 

x² = 64

 

x = 8

 

 

To get the volume, we multiply the three dimensions together.  Since each dimension is equal to each other, we can simply cube the result that was obtained.

 

V = x³

V = 8³

V = 512

 

This is close to 500 cm³.