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 f

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 $520 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.]
left border ft
bottom border ft
garden area ft2

Answer 1

triangle, maximum area? 1,690 ft^2 bottom border=130 feet, left border= 26 feet

$520 budget, $2 per foot on one side (bottom border), $10 per foot on 2nd side (left border) $0 for 3rd side (using stream as the border)

520 = 10x+2y

2y = 520-10x

y = 260-5x

A = XY/2 = x(260-5x)/2 = 130x -2.5x^2

A'= 130 -5x = 0

x = 130/5 = 26 feet = left side

y = (520-10(26))/2 = 260-130 = 130 feet = bottom side

A = XY/2 = 26(130)/2 = 13(130) = 1690 ft^2 = max Area

stream side of triangle = 26sqr26 = about 132.57 feet= the hypotenuse of the right triangle