Question

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

Answer 1

`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`

Answer 2

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.