You can
put this solution on YOUR website! The bottom of the box and all four sides are constructed of glass costing $1/ft2,
and
the top is to be constructed of glass costing $5/ft2.
Sally has $72 to spend on the box.
How should she choose the dimensions so that the volume of the box is greatest?
:
Let x = the side of the square base and top
Let h = height of the box
5 of the sides cost $1 a sq/ft
Top cost 5x^2
:
We need to get h in terms of x
:
Top + bottom + 4 sides = 72
5x^2 + x^2 + 4(xh) = 72
6x^2 + 4(xh) = 72
4xh = 72 - 6x^2
h =
Simplify
h =
:
Find the volume:
V = x^2*h
replace h
V = x^2*
Cancel x
V = x*
V =
Write the equation as:
-1.5x^3 + 18x = 0
:
Graph this
Max volume when x=2
:
find h, when x=2
h =
h =
h =
h = 6
;
A box: 2 * 2 * 6 = 24 cu (as seen on the graph)
;
Check the cost;
Top + bottom + 4 sides = $72
5(4) + 4 + 4(2*6) =
20 + 4 + 48 = $72
