The length of a rectangle is 3 times its width. If the area of the rectangle is #"192 in"^2#, how do you find its perimeter?

1 Answer
Mar 16, 2018

The perimeter is #64# inches

Explanation:

First find the lengths of the sides of the rectangle

Use the information about #area# to find the lengths of the sides.

Begin by finding a way to describe each side using math language.

Let #x# represent the width of the rectangle

Width . . . . . . . . . #x# #larr# width
#3# times that . . . #3x# #larr# length

The area is the product of these two sides
[ width ] #xx# [ length ] #=# Area
[ . . #x#. . .] #xx# [ . . #3x# . .] # =  192#

#192 = (x)(3x) #   Solve for #x#, already defined as the width

1) Clear the parentheses by distributing the #x#
#192 = 3 x^2#

2) Divide both sides by #3# to isolate #x^2#
#64 = x^2#

3) Take the square roots of both sides
#sqrt64 = sqrtx^2#

#+-8 = x#, already defined as the width of the rectangle

The width cannot be a negative number, so #-8# is a discarded solution.

Answer:
The width of the rectangle is #8# inches
So the length must be #3xx8#, which is #24# inches.

Now use the lengths of the sides of the rectangle to find its perimeter

Perimeter is the sum of all four sides

[ #2# widths ] #+ [  2# lengths ]#=# Perimeter
[... ..#2(8) ...] + [ ..2(24)..] = #Perimeter

1) Clear the parentheses
#16 + 48 =# Perimeter

2) Add
#64 =# Perimeter

Check

1) The sides should multiply up to an area of #192  "in"^2#
#8 xx 24 = 192#

#Check#