Optimization question

Question

You are building a right-angled triangular flower garden along a stream as shown in the figure. 

The fencing of the left border costs $10 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $600 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose? [The area of a right-triangle is given by A = xy/2.]

What are the dimensions of the left border, bottom border and the garden area?

Answer 1

Assuming x is the left border, then the cost = 10x + 2y = 600 or y = 300 - 5x

Plugging that into the area equation A = 1/2xy we get A = 1/2x(300 - 5x) or A(x) = 150x -5/2x²

To maximixe the area, take the derivative and set it equal to zero. A'(x) = 150 - 5x = 0 so x = 30 ft. plugging that into the y = 300 - 5x equation, we get y = 150 ft and the area = 1/2(30)(150) = 2250 sq ft