Question

A rectangular park measuring 32 yards by 24 yards is surrounded by a trail of uniform width. If the area of the park and the trail combine is 1748 square yards, what is the width of the park

Answer 1

The width = 38 yard

Explanation:

Given

Dimension of Park = 32 by 24 yard

Area = 1748 yd²

Required

Find the width of the park

Given that the park is surrounded by a trail;

Let the distance between the park and the trail be represented with y;

Such that, the dimension of the park becomes (32 + y + y) by (24 + y + y) because it is surrounded on all sides

Area of rectangle is calculated as thus;

Area = Length * Width

Substitute 1748 for area; 32 + 2y and 24 + 2y for length and width

The formula becomes

`1748 = (32 + 2y) * (24 +2y)`

Open Bracket

`1748 = 32(24 + 2y) + 2y(24 + 2y)`

`1748 = 768 + 64y + 48y + 4y^2`

`1748 = 768 + 112y + 4y^2`

Subtract 1748 from both sides

`1748 -1748 = 768 -1748 + 112y + 4y^2`

`0 = 768 -1748 + 112y + 4y^2`

`0 = -980 + 112y + 4y^2`

Rearrange

`4y^2 + 112y -980 = 0`

Divide through by 4

`y^2 + 28y - 245 = 0`

Expand

`y^2 + 35y -7y - 245 = 0`

Factorize

`y(y+35) - 7(y + 35) = 0`

`(y-7)(y+35) = 0`

Split the above into two

`y - 7 = 0\ or\ y + 35 = 0`

`y = 7\ or\ y = -35`

But y can't be less than 0;

`So,\ y = 7`

Recall that the dimension of the park is 32 + 2y by 24 + 2y

So, the dimension becomes 32 + 2*7 by 24 + 2*7

Dimension = 32 + 14 yard by 24 + 14 yard

Dimension = 46 yard by 38 yard

Hence, the width = 38 yard