Answer:
the sphere should be constructed with a radius R= 0.075 ft in order to maximise the profit
Step-by-step explanation:
since the profit function P is
P= $30/ft²* A - $800/ft³ * V = a*A + b*V
where A and V are the area and the volume of the sphere respectively.Then
A= 4*π*R²
and
V= 4/3*π*R³
where R is the radius
replacing in P
P= a*A + b*V = 4*π*a* R² - 4/3*π*b*R³ =
the maximum is found where the derivative of P with respect to R is equal to 0 , therefore:
dP/dR= 8*π*a* R - 4*π*b*R² = 0
then
8*π*a* R - 4*π*b*R² = 0
4*π*R*(2*a - b*R) = 0
since R>0
2*a - b*R=0
R= 2*a/b
replacing values
R= 2*a/b = 2*$30/ft² / $800/ft³ = 0.075 ft
thus the sphere should be constructed with a radius R= 0.075 ft in order to maximise the profit.
