Question

someone help me with a word problem!!

Answer 1

the design of a digital box camera maximized the volume while keeping the sum of the dimensions at inches. if the length must be 1.5 times the height, what should each dimension be?
hint: let x represent one of the dimensions, and then define the other dimensions in terms of x

Answer 2

will give medal!

Answer 3

L = 1.5 H = (3/2) H
(3/2)H + H + W = 6 so
W = 6 - H - (3/2)H = 6 - (5/2) H

V = L*H*W = (3/2) H * H * (6 - (5/2) H)

volume is now a function of H alone and we can maximize it by
taking the derivative. To make it easier, replace H by x.

V = (3/2) x * x * (6 - (5/2) x) = 9 x^2 - (15/4) x^3
dV/dx = 18 x - (45/4) x^2 = 0
this has the solution x = 0
obtained by factoring out x.
Then you have 18 - (45/4)x = 0
x = 72/45 as the other root.

You can calculate the inflection points or simply graph out the function to be sure whether
the points are minima or maxima. In this case, H = 72/45 = 1.6 is a maximum giving for the
dimensions:
H = 1.6
L = 2.4
W = 2.0

and the volume V = 7.68