a farm silo is to be built which is a hemispherical dome ato

a farm silo is to be built which is a hemispherical dome atop a cylinder. the total amount of money available for materials is $10450. The brick cylinder floor costs$11.25 per square foot. The metal cylinder sides cost $6.45 per square foot. The hemispherical wooden dome costs $8.75 per square foot. What are the dimensions and volume of the silo of largest total volume which can be built?

Solution

Let r be the radius of cylinder and hemi sphere and length of

cylinder be h.

Curved area of hemisphere = 2pi*r^2  

curved surface of cylinder = 2pi*r*h

base area of cylinder = pi*r^2

Total money available = 10450

So, 11.25(pir^2) + 6.45(2pi*r*h) + 8.75*(2pi*r^2) = 10450

90.32r^2 + 40.52rh = 10450

h = ( 1045 - 90.32r^2)/4.052r

Volume = (2pi*r^3/3) + pi*r^2*h

V(r) = 2pi*r^3/3 + pi*r^2*( ( 1045 - 90.32r^2)/4.052r

= 2pi*r^3/3 + pi*r(1045 - 90.32)/4.05

find dV/dr =0 and solve for r