Answer:
side of 6.124 ft and height of 3.674 ft
Step-by-step explanation:
Let's s be the side of the square base and h be the height of the rectangular box.
The base and the roof would have an area of [tex]s^2[/tex] and cost of
[tex]4s^2 + 2s^2 = 6s^2[/tex]
The sides would have an area of 4sh and cost of 4sh*2.5 = 10sh
So the total cost for the material is
[tex]6s^2 + 10sh = 450[/tex]
[tex]10sh = 450 - 6s^2[/tex]
[tex]h = 45/s - 0.6s[/tex]
The volume of the shed has the following formula
[tex]V = s^2h = s^2(45/s - 0.6s) = 45s - 0.6s^3[/tex]
To find the maximum value for V, we can take its first derivative, and set it to 0
[tex]\frac{dV}{ds} = 45 - 1.2s^2 = 0[/tex]
[tex]1.2s^2 = 45[/tex]
[tex]s^2 = 45 / 1.2 = 37.5[/tex]
[tex]s = \sqrt{37.5} = 6.124 ft[/tex]
h = 45/s - 0.6s = 3.674 ft
