the area of a rectangle is 108 square feet and the perimeter is 42 feet find the dimension of the rectangle

Question

i need help with math word problems a lot. i have a lot of trouble with these.

Answer 1

So we have Area = 108 and perimeter = 42

we know that the equation for area of a rectangle is Area = Length * width

which means 108 = L*W

we also know the equation of the perimeter is Perimeter = length +width + length + width

which means 42 = 2L + 2W

Solve for either L or W

in this case we will solve for w which is W = 21 - L

plug that in to your equation for area.

108 = (21-L)L

Then distribute

108 = 21L - L^2

move everything until one side.

L^2 - 21L + 108 = 0

factor

(L - 9)(L - 12) = 0

so L = 9 or L = 12

solve for W = 21 - L

W = 12 or W = 9

so the rectangle is a 9X12 rectangle.

Answer 2

Hi Jesus,

 

the dimensions are 12 by 9.  Here is how:

 

A = L * W

 

A = 108 = L * W

108 = L * W

 

 

perimeter = 2l + 2w  = 2 (l + w)

 

perimeter = 2 (l + w)    42 = 2 (l + w)   21 = (l + w)   21 = L + W

 

solve for either L or W. Let's solve for L:  L = 21 - W

 

 

Go back to area equation and substitute: 

 

A = L * W   A = (21 - W) (W)

 

108 = (21 - W) (W) = 21W - W²

 

108 = 21W - W2    Move (21W - W2) to the left side

 

W2 - 21W + 108 = 0

 

Use quadratic formula (if you are taking Algebra 1) to get the values:

 

 

a = 1, b= -21, c = 108

 

21 + (441 -432)^1/2

-----------------------                          

    2

 

(I re-wrote (1/2) power notation for square root sign)

 

 

21 + (9)1/2            21 + 3              24         18

--------------   =   -----------    =    -----   ,   ------    =   12, 9

    2                            2                 2            2

 

In conclusion, dimensions are 12 ft by 9 ft.

 

For perimeter, 2 (L + W) = 2 (12 + 9) = 2 (21) = 42 ft   (It works)

For area, L * W = 12 * 9 = 108 sq ft   (It works)

 

There you have it.  Good luck.

 

Yohan Chu   (LV  NV)