The perimeter of a rectangular driveway is 68 feet. The area is 280 square feet. What are the dimensions of the driveway?

2 Answers
Nov 21, 2016

#1)w=20ft, l=14ft#
#2)w=14ft, l=20ft#

Explanation:

Let's define the variables:

#P: #perimeter
#A:# area
#l: #length
#w:# width

#P=2l+2w=68#

Simplify (divide by #2#)

#l+w=34#

Solve for #l#

#l=34-w#

#A=l*w=280#

Substitute #34-w# instead of #l#

#A=(34-w)w=280#

#-w^2+34w=280#
#-w^2+34w-280=0#

Multiply by #-1#

#w^2-34w+280=0#

Factorize

#(w-20)(w-14)=0#

Set each expression equal to zero

#1)w-20=0#
#w=20#

#2)w-14=0#
#w=14#

Option #1#) substitute #20# instead of #w#

#l+w=34#
#l+20=34#
#l=14#

Option#2#) substitute #14# instead of #w#

#l+w=34#
#l+14=34#
#l=20#

#1)w=20ft, l=14ft#
#2)w=14ft, l=20ft#

Nov 21, 2016

The dimensions are #20# and #14# feet. See explanation.

Explanation:

We are looking for the dimensions of a rectangle, so we are looking for 2 numbers #a# and #b# which satisfy the set of equations:

#{(2a+2b=68),(a*b=280):}#

To solve this set we calculate #b# from the first equation:

#a+b=34 => b=34-a#

Now we substitute #b# in the second equation:

#a*(34-a)=280#

#34a-a^2=280#

#-a^2+34a-280=0#

#Delta=1156-1120=36#

#sqrt(Delta)=6#

#a_1=(-34-6)/(-2)=20#

#a_2=(-34+6)/(-2)=14#

Now we have to calculate #b# for each calculated value of #a#

#b_1=34-a_1=34-20=14#

#b_2=34-a_2=34-14=20#

So we see that the dimensions are #20# and #14# feet.