A silo is essentially a cylinder with a half sphere on top; let the cylinder have height h and let the cylinder and sphere have radius r. What is the cost of building a silo of volume 1000 cubic meters. The curved surface on top of the silo costs $3 per square meter to build, while the walls cost $1 per square meter.
a. Find functions for the volume and the surface area of the silo.
b. Using the Lagrange multiplier method, find the dimensions such that the surface area is minimized for the given volume. In this case, F is the surface area function.
c. Do these dimensions for part b seem reasonable? Why or why not?
d. Set up equations for the volume and the cost of building the silo.
e. Using the Lagrange multiplier method, find the cheapest way to build the silo.
f. Do these dimensions for part c seem reasonable? Why or why not?
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