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 $8 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $480 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

You are building a right-angled triangular flower garden along a stream as shown in the figure. The fencing of the left border costs $8 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $480 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

Consider x be the left side and y be the bottom side of the right-angle triangular garden. Consider the required figure as shown below The left border cost is S8, and the lower border cost is $2. The total spend is $480. Therefore 8x 2y 480 2y 480-8r 480-8x 1V 2 y = 240-4x The area of the right-angle triangle is, Ay x(240-4x) A =x(120-2x) = 120x-2x2

dA Solve0 to find the value of x cr dA cr 120-4x = 0 4x 1 20 x=30ft Second derivative test: Consider f(x) is a function of x which is twice differentiable at xo 1. If f" (xo)>0 then, fhas a local minimum at xo 2. If f"(xo) <0 then,/has a local maximum at ao dA with respect to x to find the maximum and minimum value. Differentiate d- A dr2 < 0 By second derivative test x 30 is the maximum value. The value of y is, y = 240-4 (30) 240-120 120 ft

The maximum area of the garden is, A- 1(80)120) 2 -1800 ft2