Rectangles?

Question

Find the dimensions of a rectangle whose area is 180 cm² and whose perimeter is 54cm. 

Answer 1

Area = (Length)(Width)
A = L*W
A = 180
Perimeter = 2L +2W
54 = 2L + 2W

A = L*W
Divide each side by W.
A / W = L*W/W
(A / W) = L
(180 / W) = L

P = 2L + 2W
P = 2(L + W)
P / 2 = [2(L+W)] / 2
P / 2 = L + W
(54 / 2) = L + W

27 = L + W

27 - W = L + W - W

27 - W = L

 

27 - W = (180 / W)

(27 - W)(W) = (W)(180 / W)

W(27 - W) = 180

27W - W²= 180

27W - W² + W² = 180 + W²

27W = W² + 180

27W - 27W = W² + 180 - 27W

0 = W² - 27W + 180

 

From here, we need to figure out what two numbers can multiply to equal 180 and add to equal (-27). We then will put those numbers in the following form: (W + __)(W + __).

 

(-15)(-12) = 180

(-15) + (-12) = -15 - 12 = -27

 

(W - 12)(W - 15) = 0

Then, we can separate them out and solve.

 

(W - 12) = 0; (W - 15) = 0

 

W - 12 = 0

W - 12 + 12 = 0 + 12

W = 12

 

W - 15 = 0

W - 15 + 15 = 0 + 15

W = 15

 

So, the dimensions of the rectangle are: 15cm and 12cm.

 

You can double check this by plugging these numbers back into the Area and Perimeter equations:

A = LW

A = (15)(12)

A = 180

180 = 180

 

P = 2(L+W)

P = 2(15 + 12)

P = 2(27)

P = 54

54 = 54