an ecology center wants to set up an experimental garden. it has 200 meters of fencing to enclose a rectangular area of 2400 square meters. find the length and width of the rectangle

Answer 1

The rectangular garden has an area of 2400 m². There are 200 m of fencing available.

The first instinct might be to divide 200 by 4 and have 50 m per side, i.e. a square. This won't work because 50 * 50 = 2500 which is 100 m² too large.

A thought experiment:

Take a look at 2400 and think what 2 numbers multiplied together have that as their product. The choices are 10, 240; 20, 120; 30, 80; 40, 60.

Now consider that the sum of 2 times each number must not exceed 200.

2x10 = 20, 2x240 = 480 20+480 = 500, so this won't work.

2x20 = 40, 2x120 = 240 40+240 = 280, so this won't work

2x30 = 60, 2x80 = 160 60+160 = 220, so this won't work

2x40 = 80, 2x60 = 120 80+120 = 200, so 40 m and 60 m are the correct dimensions.

Mathematically, set up the following equations:

(1) L * W = 2400

(2) 2L+2W = 200

Solve (2) for L in terms of W:

(2a) L = 100 - W (divided all terms by 2 then subtracted W from both sides)

Replace L in (1) by 100-W

(1a) (100-W)*W = 2400

100W - W² = 2400

multiply all terms by -1 to get W² - 100W = -2400

add 2400 to both sides:

W² - 100W + 2400 = 0

Solve for 2 solutions to W using the quadratic formula (-b +/- sqrt(b² - 4ac))/2a, where a = 1, b = -100, c = 2400

W = (-(-100) +/- sqrt((-100²) - 4(1)(2400)) / 2(1)) = (100 +/- sqrt(10000 - 4*2400))/2

W = (100 +/- 20) / 2

W = 120/2 = 60 AND W = 80/2 = 40, same as before.