Finding the surface area of a rectangular prism given its volume and side length ratios.

Question

A rectangular prism has a width that is equal to its length. Its height is twice as long as its width. The volume of the rectangular prism is 250. What is the surface area of the rectangular prism?

Answer 1

1. First, let's first define our variables

Let V = volume

Let SA = surface area

Let l = length

Let h = height

Let w = width

2. What does the problem tell us about the ratios of the height, length, and width? Let's write math to express these ratios.

a. The length and the width are the same.

l= w

b. The height is twice as long as the width.

h = 2w

3. What is our equation for volume?

The volume of an object represents its 3-D space measured cubic units. This means it's essentially how much can fit in this object. For a rectangular prism, you can think of the space fitting inside its base area in layers to reach its height, like in a tissue box. This can be represented as the (area of the base) × height, which we can write as follows:

V = l × w × h

4. Let us rewrite our equation for volume since we know the ratios of h, l, and width:

V = l × w × h

l= w

h= 2w

Substitute those in:

V = w × w × 2w

Now combine those algebraically:

V = 2w³

5. Let's solve for w. We also know that V = 250.

V= 2w³

2w³ = 250

Divide both sides by 2:

w³ = 125

Take the cube root of both sides

w = ³√(125)

w = 5

6. Now we can solve for the other 2 sides:

l = w, w=5, so l=5

h = 2w = 2*5 = 10

7. How do we find the surface area?

1. If you think of a rectangular box, there are 6 surfaces. If you picture it lying on the ground in front of you, there are several sides (1) touching the ground (2) facing the ceiling (3) facing toward you (4) facing away from you (5) facing left (6) facing right.
2. The length is the horizontal box edge on the ground going from left to right in front of you. The width is the horizontal box edge on the ground that starts in front of you and goes away from your face. The height is the vertical box edge that starts at the floor and goes to the top of the box.
3. Sides 1 and 2: If the base is lying on the ground, that's formed by the length and the width. This is the same area as that facing the ceiling. These two sides each have an area of l*w. The total area of sides 1 and 2 is 2lw
4. Sides 3 and 4: The sides facing toward and away from you are both formed by the length and the height. These two sides each have an area of l*h. The total area of sides 3 and 4 is 2lh.
5. Sides 5 and 6: The sides facing left and right are formed by the width and the height. These two sides each have an area of w*h. The total area of sides 5 and 6 is 2wh.

The total surface area of the prism is found by adding the surface area of all 6 sides:

SA = 2lw + 2lh + 2wh

8. Now let's substitute what we solved for l, w, and h into our equation for surface area:

w = 5

l = 5

h = 10

SA = 2lw + 2lh + 2wh = 2×5×5 + 2×5×10 + 2×5×10 = 10×5 + 10×10 + 10×10 = 50 + 100 + 100 = 250

So the surface area for this rectangular prism is also 250!