SOLUTION: 2. A crate has a width of 8 feet. The areas of all 6 sides of the crate total 42 ft2 feet. Express the height h of the crate in terms of the length x (i.e. write h as a function

Algebra ->  Volume -> SOLUTION: 2. A crate has a width of 8 feet. The areas of all 6 sides of the crate total 42 ft2 feet. Express the height h of the crate in terms of the length x (i.e. write h as a function       Log On


   

Question 1127062: 2. A crate has a width of 8 feet. The areas of all 6 sides of the crate total 42 ft2
feet. Express
the height h of the crate in terms of the length x (i.e. write h as a function of x ). Simplify
your answer.
a) Express the volume V of the box as a function of the length “l”.
b) Find the maximum volume that such a box could have.
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
8, width
h, height
x, length
42, surface area all six sides

2hx%2B2%2A8h%2B2%2A8x=42
hx%2B8h%2B8x=21
-
Height as Function of Length
h%28x%2B8%29%2B8x=21
h%28x%2B8%29=21-8x
highlight_green%28h=%2821-8x%29%2F%28x%2B8%29%29

Volume Formula
v=8x%28%2821-8x%29%2F%28x%2B8%29%29

v=%28168x-64x%5E2%29%2F%28x%2B8%29
-
dv%2Fdx=%28%28x%2B8%29%28168-128x%29-%28168x-64x%5E2%29%2A1%29%2F%28x%2B8%29%5E2
skipping all the steps,
finding extreme value of v
%28-64x%5E2-1024x%2B1344%29%2F%28x%2B8%29%5E2=0

highlight_green%28x%5E2%2B16x-21=0%29


x=%28-16%2Bsqrt%284%2A4%2A16%2B4%2A21%29%29%2F2

highlight_green%28x=-8%2Bsqrt%2885%29%29%29
If no mistakes made, this is the length, x, for maximum volume v.