The length of a box is 2 centimeters less than its height. The width is 8 centimeters more. Write a polynomial that represents the volume of the box.

Question

If the box has a volume of 96 cubic centimeters, what is the surface area?

Answer 1

We know the following:

 

Volume = length × width × height

 

 

We need to assign variables for length, width, and height.  Since the height is being compared to both quantities,

 

Let   height (h) = x

       length (l) = x - 2

       width (w) = x + 8

 

Volume = x(x - 2)(x + 8)

           = x(x² + 6x - 16)

           = x³ + 6x² - 16x

 

 

To find the surface area, we need to add up the flat areas of the box.  This requires us to find the three dimensions.  We know the volume.

 

x³ + 6x² - 16x = 96

 

 

Subtract 96 on both sides of equation.

 

x³ + 6x² - 16x - 96 = 0

 

x²(x + 6) - 16(x + 6) = 0

 

(x² - 16)(x + 6) = 0

 

(x + 4)(x - 4)(x + 6) = 0

 

x = -4         x = 4        x = -6

 

 

We have 3 solutions of x.  Since we cannot have negative values for dimensions, we accept the positive value of x.

 

x = 4

 

 

Substitute this value of x into the dimensions.

 

height (h) = 4 cm

length (l) = 2 cm

width (w) = 12 cm

 

Now we can find the surface area.  We must add all the flat areas of the box.  If we draw a box and label it with h, l, and w, we can establish a formula. 

 

SA = 2(base) + 2(front) + 2(side)

 

SA = 2lw + 2lh + 2wh

 

SA = 2(2)(12) + 2(2)(4) + 2(12)(4)

 

     = 48 + 16 + 96

     =160 cm²

 

The surface area is 160 cm².