Algebra 2 Optimization Problem - How much fencing do you need?

Question

You are designing a rectangular garden. The garden will be enclosed by fencing on three sides and by a house on the fourth side. You want the garden to have an area of 288 square feet, but you want to minimize the amount of fencing that you need. How many feet of fencing do you need to enclose the garden?

Answer 1

Let x = length of side parallel to the house

     y = length of each of the other two sides

 

Given:  xy = 288  So, y = 288/x

 

Length of fence = L = x + 2y

                             = x + 576/x

 

L' = 1 - 576/x² = (x²-576)/x²

 

L' = 0 when x = 24

 

If 0 < x < 24, then L' < 0  so, L is decreasing

 

If x > 24, then L' > 0  so, L is increasing

 

Minimum length when x = 24 ft and y = 288/24 = 12 ft

 

Note:  I used Calculus to solve the problem.  If you don't know Calculus, another way to do the problem would be to use a graphing calculator to graph the length function and see that the minimum length occurs when x = 24.  

 

 

Answer 2

Perimeter = L + 2W

 

Area = L*w

 

288 = L*w

 

 

288/L = w

 

Perimeter = L + 2(288/L)

= L + 576/L Graphing the perimeter function P(L) = (L + 576)/L at www.desmos.com, the minimum point is (24,48)