Real-life Calculus: Maximum Volume

Question

The surface area is 337.5 of a box with a square base. I am using

S = x² + 4xh = 337.5

then I get 

(337.5 - x²)/4x

So

V = x²(337.5 - x^2)/(4x)

After finding f'(x) = 84.375 - (3x²)/(4) and a lot of messy math because of that stupid decimal, I come up with a long decimal answer that I'm sure is not correct. What am I doing wrong here?

 

The answer I came up with was h = 5.59 and x = 10.606

Answer 1

Adam,

 

Here is what I get:

 

S = x^2 +4xh = 337.5

 

Rearrange to solve for h.

 

h = (337.5 - x^2)/(4x) = 337.5/(4x) - (1/4)(x)

 

V = x^2h

V = x^2[337.5/(4x) - (1/4)(x)]

V = (337.5/4)x - (x^3)/4

 

Take dV/dx and set = 0:

 

0 = 337.5/4 - (3x^2)/4

 

Simplify:

 

337.5 -3x^2 = 0

 

x^2 = 112.5

 

x = +10.61 or -10.61.

 

Take only positive solution.

 

x = 10.61   (used 4 sig. figs. since 4 are initially given in the problem)

h =  (337.5-(10.61^2))/((4*10.61)) = 5.300  (again used 4 sig. figs. since 4 are initially given). 

 

 

Hope this helps!

 

Art B.