Question

A 1590 cubic foot open-top rectangular tank with a square base x feet on a side and y feet deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy. If the total cost is

c=4(x²+9xy)+30xy, what values of x and y minimize the cost?

Answer 1

The question is a calculus optimization problem where one must minimize the total cost function c=4(x²+9xy)+30xy for a tank with a volume of 1590 cubic feet, given a square base with side x and depth y. Calculus techniques such as derivatives and possibly Lagrange multipliers are used to find the minimum values of x and y that satisfy the volume constraint.

Step-by-step explanation:

The question involves finding the minimum total cost for building a tank, where the cost function is given by c=4(x²+9xy)+30xy. Here, x and y are the sides of the square base and the depth of the tank, respectively. The volume of the tank is given as 1590 cubic feet, so we have the volume constraint x²y=1590.To minimize the cost function, we need to use calculus to find the values of x and y that minimize the cost function subject to the volume constraint. This involves setting up a system of equations, one for the cost function and another for the volume constraint, and solving for x and y using Lagrange multipliers or other optimization techniques such as substitution from the constraint into the cost function.

Since this is a typical optimization problem in calculus, it would usually be solved in detail by obtaining the first derivatives of the cost function with respect to both x and y, setting them to zero, and solving the resulting system of equations, possibly with the help of the volume constraint.