Question is in the description

Question

A box is a rectangular prism with a volume of 768 cubic inches. The length of the box is 4 inches more than its height. It’s width is the difference of 16 and it’s height.

a) Draw a diagram of the box and label it’s dimensions in terms of its height.

b) Write a polynomial that represents the volume of the box.

c) Use the polynomial from part b to find two sets of possible dimensions of the box.

d) Which set of dimensions results in a box with the least possible surface area? Explain your reasoning.

Answer 1

L=4+H

W=16-H

Volume = LWH = 768 = (4+H)(16-H)H = (64+12H-H^2)H = -H^3 +12H^2 +64H = 768

H^3 -12H^2 - 64H + 768 = 0

It's difficult to solve a cubic equation, but try a few easy integrals.

768, 64 and 12 all are divisible by 4, try 4 or 8

H=8 works. Make it a factor, H-8. Divide the cubic by H-8

to get a quadratic. The quadratic factors into (H+8)(H-12)

The polynomial has 3 factors

(H+8)(H-8)(H-12) = 0

with solutions: H= -8, +8 and +12. Ignore the negative solution, leaving

H= 8 or 12

Surface Area is 2 times top & bottom + sometimes 4 times the sides

The box has width= 16-H, length 4+H

If H = 8, L=4+8=12 and W =16-8= 8

It's 12x8x8 = 768

Or if H=12, then L= 4+12=16 and W=16-12=4

It's 12x16x4 = 768

Those are the two solutions,

12x16x4 or 12 x8x8

surface area is 2 times the top&bottom and 4 times a side

There are several possibilities,

8x8 for the top&bottom would be 64x2=32 + the 4 sides 4(8x12) = 32 + 4(96) = 32 + 384 = 416

that could be lowest

But there're other possibilities with 12x16x4

top & bottom with 4x12 = 2(48)=96

sides are 12x16 and 16x4 = 192 and 64

192+64+96 = 352 < 416 with the other dimensions

That might be the lowest surface area=352, probably is

Answer 2

h*(h+4)*(h-16)=768

h^2+4h(h-16)=768

h^2+4h^2-64h=768

5h^2-64h=768

h=20.3485

l=24.348

w=4.3485