finding dimenaions and area

Question

You are building a right-angled triangular flower garden along a stream as shown in the figure. (The borders can be in any direction as long as they are at right angles as shown.)

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 $720 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose?

left border____ ft

bottom border____ ft

garden area____ ft²

Answer 1

x is the left border.

Area = .5xy

10x + 2y = 720

2y = 720 - 10x

y = 360 - 5x Substitute that into the Area equation.

Area = .5x(360-5x) = 180x - 2.5x^2

That's an upside down parabola. Find where the derivative = 0. That's the vertex, the max.

A' = 180 - 5x = 0

180 = 5x

36 = x

y = 360 - 5x = 360 - 5*36 = 360 - 180 = 180

Left border is 36 ft.

Bottom border is 180 ft.

Area is 36*180 = 6480 ft^2