posted October 30, 2007 15:32
Does anyone know how to attempt these questions:
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 8400 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize the cost. _______________________________________________
An offshore oil rig well is 4 kilometers off the coast. The refinery is 9 kilometers down the coast. Laying pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost? Complete the following statement.
The pipe should be laid in the ocean from the rig to a point on the coast____km from the refinery, then on land the remainder of the distance to the refinery.
posted October 30, 2007 19:23
For the first problem, you can start by defining the volume in terms of R (the radis of the caps and cylinder) and L (the length of the cylinder without the caps). This gives the identity 8400 = 4/3 * PI * R^3 + PI * R^2 *L. You can rewrite that to define L in terms of R. For the total surface cost you have COST = 2 * PI * R * L + 2 * (SURFACE OF A SPHERE GIVEN R (which you can look up in your textbook)). Then substitute L in terms of R from the first equation into the second and minimize that.
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