Question

Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area?

Answer 1

The possible widths of the play area are : 30 ft. or 60 ft.

Explanation:

Let length be `l` and width be `w`

Perimeter = `180 ft. = 2(l+ w)`---------(1)

and

Area = `1800 ft.^2 = l xx w`----------(2)

From (1),

`2l+2w = 180`

`=> 2l = 180-2w`

`=> l = (180 - 2w)/2`

`=> l = 90- w`

Substitute this value of `l` in (2),

`1800 = (90-w) xx w`

`=> 1800 = 90w - w^2`

`=> w^2 -90w + 1800 = 0`

Solving this quadratic equation we have :

`=> w^2 -30w -60w + 1800 = 0`

`=> w(w -30) -60 (w- 30) = 0`

`=> (w-30)(w-60)= 0`

`therefore w= 30 or w=60`

The possible widths of the play area are : 30 ft. or 60 ft.

Answer 2

`30" or "60" feet"`

Explanation:

`"using the following formulae related to rectangles"`

`"where "l" is the length and "w" the width"`

`• " perimeter (P) "=2l+2w`

`• " area (A) "=lxxw=lw`

`"the perimeter will be "180" feet "larrcolor(blue)"fencing"`

`"obtaining "l" in terms of "w`

`rArr2l+2w=180`

`rArr2l=180-2w`

`rArrl=1/2(180-2w)=90-w`

`A=lw=w(90-w)=1800`

`rArrw^2-90w+1800=0larrcolor(blue)"quadratic equation"`

`"the factors of + 1800 which sum to - 90 are - 30 and - 60"`

`rArr(w-30)(w-60)=0`

`"equate each factor to zero and solve for "w`

`w-30=0rArrw=30`

`w-60=0rArrw=60`