Question

Kiana has a rectangular back yard that is 50 feet wide and 60 feet deep. She plans to construct a pool area and a patio area as shown in Figure 1. She can spend at most $10,500 on the project. The patio area must be at least as large as the pool area, while staying within the back yard area. The pool area will cost $5 per square foot, the patio will cost $3 per square foot, and the pool area should be at least 500 ft2.

When formulating a Linear Program (LP) to determine the depth of the pool and the depth of the patio while mini- mizing the total back yard area unused, Kiana defined the following Decision Variables:

x: depth (ft) of the patio area, y: depth (ft) of the pool area.

Find the constraints and any redundant constraints

find the optimal solution

Answer 1

The depth of patio is and of pool is . So, their respective areas are and . The patio area at least as large as pool area says . The cost for pool area is and that of patio is . Pool area is at least sq ft, so . Finally, . Total cost is . Total backyard area unused is .

Thus, the LP is

Simplifying the constraints gives

We solve this using graphical method. The graph of the constraints

is as follows:

Thus, the corner points are , , , and . At these points, the value of the objective function, is , , , and . Thus, the minimum is at , and it is .

Thus, the optimal solution is: , . Patio is ft deep, pool is ft deep.