Question

Aaron has 42 m of fencing to build a three-sided fence around a rectangular plot of land that sits on a riverbank. (the fourth side of the enclosure would be the river. ) the area of the land is 216 square meters. List each set of possible dimensions (length and width) of the field.

Answer 1

Two possible sets of dimensions:

Width (m) Length (m)

1) 12 18

2) 9 24

Explanation:

Fence perimeter (3 sides) = 42 m

Area = 216 m^2

Let W and L be the width and length. Assume L to be the length of the side that is opposite the riverbank. The perimeter needs only 1 L, instead of 2.

P = 2W + L

42 m^2 = 2W + L

L = 42-2W [Rearrance to isolate one of the two variables]

L*W = 216 [The area, 216 m^2, is equalt to the Length*Width]

(42-2W)*W = 216 [Sunstitute the expression for L (42-2W)]

42W-2W^2 = 216

-2W^2 + 42W -216 = 0

W^2 -26W + 108 = 0 [Rearrange to form a quadratic equation]

(W-12)(W-9) = 0 [Factor, or solve the quadratic equation]

W = 12 or 9 [There are two possible values for W]

Assume W, then calculate L from W*L=216 ft^2

W=12 m, then L= 18 m

W=9 m, then L = 24 m

Check:

A) Do these dimensions result in 216 m^2?

a. (12)*(18) = 216 YES

b. (9)*(24) = 216 YES

B) does the 3-sided perimeter add to 42 m?

P = 2W + L

a. 42 = 2(12) + 18? YES

b. 42 = 2(9) + 24? YES

These 2 sets of dimensions satisfy all the constraints.

Width (m) Length (m)

1) 12 18

2) 9 24