Answer:
Width of walkway = 3.71625 feet
Step-by-step explanation:
Let the width of the walkway be w. Then the length of entire area of the pool including the walkway is 32 + 2w and the breadth of the entire walkway is 16 + 2w since there is a width of w on both sides of length and breadth
Total Area of pool with pathway
(16+ 2w)(32+2w) = 924
Using the FOIL method we can expand the term on the left as follows:
= [tex]\sf 16\cdot \:32+16\cdot \:2w+2w\cdot \:32+2w\cdot \:2w[/tex]
= [tex]\sf 512+96w+4w^2[/tex]
Rearrange terms to get
[tex]\sf 4w^2 + 96w + 512[/tex]
So we get
[tex]\sf 4w^2 + 96w + 512 = 924[/tex]
Subtract 924 from both sides
[tex]\sf 4w^2 + 96w + 512 - 924 = 0[/tex]
==> [tex]\sf 4w^2 + 96w -412 = 0[/tex]
This is a quadratic equation of the form [tex]\sf ax^2 + bx + c[/tex] whose roots(solutions) are
[tex]\displaystyle \sf x_{1,\:2}=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Here a = 4, b = 96 and c = -412
Plugging in these values we get
[tex]\sf w_{1,\:2}=\dfrac{-96\pm \sqrt{96^2-4\cdot \:4\left(-412\right)}}{2\cdot \:4}[/tex]
[tex]\sf \sqrt{96^2-4\cdot \:4\left(-412\right)}\\\\ = \sqrt{96^2+4\cdot \:4\cdot \:412} \\\\= \sqrt{96^2+6592} \\\\= \sqrt{9216+6592} \\\\= \sqrt{15808} = 125.73\\\\[/tex]
So
[tex]w_{1,\:2}=\dfrac{-96\pm \:125.73}{2\cdot \:4}[/tex]
[tex]w_1=\dfrac{-96+125.73}{2\cdot \:4},\:w_2=\dfrac{-96-125.73}{2\cdot \:4}[/tex]
We can ignore w₂ since it is a negative value
So
[tex]\sf w = \dfrac{-96 + 125.73}{8} = 3.71625\; feet[/tex]
[tex]\boxed{ \mathsf{Width\; of\; walkway = 3.71625\;feet}}[/tex]
