Constraint Equation.

Question

A rectangular garden of area 532 square feet is to be surrounded on three sides by a brick wall costing

$ 14 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden such that the cost of the materials is minimized.

Eliminate L from the following objective equation using the above constraint equation.

1. C = 14 ( W+2L ) + 5W
2. C(W) = 19( W + 784/W )

This is a step in solving my problem in my textbook. However I am confused as to how 784 came into the equation. What was multiplied to get it?

Answer 1

Hi Amanda,

It would have been easier to see if the book had shown an intermediate step.

The key is that LW = 532, so L = 532/W.

C(W) before substitution results in C(W) = 14W + 28L + 5W = 19W + 28L.

Or, C(W) = 19W + 28(532/W). The key to getting to the equation showing in the book is that 19 divides 532 evenly, so this could be written as C(W) = 19W + 28(19)(28)/W. Then the "book" factors out the 19, giving C(W) = 19 (W + 28(28)/W), But 28(28) = 784, so that is where the 784 came from.