Question

Jessie is fencing in a rectangular plot outside of her back door so that she can let her dogs out to play. She has 60 feet of fencing and only needs to place it on three sides of the rectangular plot because the fourth side will be bound by her house. What dimensions should Jesse use for the plot so that the maximum area is enclosed? What is the Maximum Area?

Answer 1

Dimensions are length 15 feet and width 30 feet.

Maximum area of the plot is 450 feet².

Explanation:

Let the length of the plot = x feet and width = y feet.

Since, the amount of fencing to be applied on the three sides of the plot is 60 feet.

We have,

`2x + y = 60` i.e.
`y=60-2x`

Now, Area of the rectangular plot = Length × Width

i.e. Area of the rectangular plot =
`x * y`

i.e. Area of the rectangular plot =
`x* (60-2x)`

i.e. Area of the rectangular plot =
`-2x^2+60x`

Since, the maximum of the quadratic equation
`ax^2+bx+c` will be obtained at
`x= (-b)/(2a)`.

Then, the maximum of the area of the plot is at the point
`x= (-60)/(2* (-2))` i.e. x= 15

Then, maximum area of the rectangular plot =
`-2x^2+60x`

i.e. maximum area of the rectangular plot =
`-2* 15^2+60* 15`

i.e. maximum area of the rectangular plot = -450+900 = 450 feet²

Hence, Maximum area of the plot is 450 feet².

Then,
`y=60-2* 15` implies
`y=60-30` i.e. y= 30.

Hence, the maximum area is enclosed by the length 15 feet and width 30 feet.