Answer
$7.12$ ft
Work Step by Step
$H^{2}=B^{2}+P^{2}$
$20^{2}=x^{2}+(x+8)^{2}$
$400=x^{2}+x^{2}+64+2(8x)$
$400=2x^{2}+16x+64$
$2x^{2}+16x+64-400=0$
$2x^{2}+16x-336=0$
$x^{2}+8x-168=0$
Now, we solve the equation:
Step 1: Comparing $x^{2}+8x-168=0$ to the standard form of a quadratic equation $ax^{2}+bx+c=0$;
$a=1$, $b=8$ and $c=-168$
Step 2: The quadratic formula is:
$x=\frac{-b \pm \sqrt {b^{2}-4ac}}{2a}$
Step 3: Substituting the values of a,b and c in the formula:
$x=\frac{-(8) \pm \sqrt {(8)^{2}-4(1)(-168))}}{2(1)}$
Step 4: $x=\frac{-(8) \pm \sqrt {64+672}}{2}$
Step 5: $x=\frac{-8 \pm \sqrt {736}}{2}$
Step 6: $x=\frac{-8 \pm \sqrt {16\times46}}{2}$
Step 7: $x=\frac{-8 \pm 4\sqrt {46}}{2}$
Step 8: $x=-4 \pm 2\sqrt {46}$
Step 9: $x=9.56$ or $x=-17.56$
As a result, we chose x=9.56 since distance must always be positive.
Therefore, distance covered by walking on the sidewalk:
$x+(x+8)=9.56+9.56+8=27.12$ ft
Distance saved= $27.12-20=7.12$ ft
