Answer:
The width = 38 yard
Step-by-step 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
[tex]1748 = (32 + 2y) * (24 +2y)[/tex]
Open Bracket
[tex]1748 = 32(24 + 2y) + 2y(24 + 2y)[/tex]
[tex]1748 = 768 + 64y + 48y + 4y^2[/tex]
[tex]1748 = 768 + 112y + 4y^2[/tex]
Subtract 1748 from both sides
[tex]1748 -1748 = 768 -1748 + 112y + 4y^2[/tex]
[tex]0 = 768 -1748 + 112y + 4y^2[/tex]
[tex]0 = -980 + 112y + 4y^2[/tex]
Rearrange
[tex]4y^2 + 112y -980 = 0[/tex]
Divide through by 4
[tex]y^2 + 28y - 245 = 0[/tex]
Expand
[tex]y^2 + 35y -7y - 245 = 0[/tex]
Factorize
[tex]y(y+35) - 7(y + 35) = 0[/tex]
[tex](y-7)(y+35) = 0[/tex]
Split the above into two
[tex]y - 7 = 0\ or\ y + 35 = 0[/tex]
[tex]y = 7\ or\ y = -35[/tex]
But y can't be less than 0;
[tex]So,\ y = 7[/tex]
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
