Explanation:
the question is about distance.
what do we use to measure distance ? things like feet, yards, meters, ...
definitely not square feet (areas) or cubic feet (volume).
it is a right-angled triangle. so Pythagoras applies
c² = a² + b²
c is the Hypotenuse (the side opposite of the 90° angle). in our case the 48 ft side.
a and b are the legs. in our case x and x+6.
the saved walking distance is the difference between 48 ft and the sum of the legs (which would be the walking distance, when they cannot get though the lot).
48² = x² + (x + 6)² = x² + x² + 12x + 36
2304 = 2x² + 12x + 36
1152 = x² + 6x + 18
x² + 6x - 1134 = 0
the solution for such a quadratic equation
ax² + bx + c = 0
is
x = (-b ± sqrt(b² - 4ac))/(2a) =
= (-6 ± sqrt(6² - 4×1×-1134))/(2×1) =
= (-6 ± sqrt(36 + 4536))/2 =
= (-6 ± sqrt(4572))/2 =
= -3 ± sqrt(4572/4) = -3 ± sqrt(1143) =
= -3 ± sqrt(9×127) = -3 ± 3×sqrt(127)
x1 = -3 + 3×sqrt(127) = 30.80828301...
x2 = -3 - 3×sqrt(127) = -36.80828301...
negative numbers for a distance don't make any sense, so
x = 30.80828301... is our solution.
that means the 2 legs are
30.80828301... ft
36.80828301... ft
in total they are
67.61656602... ft
and so they save
67.61656602... - 48 = 19.61656602... ft ≈
≈ 19.617 ft
by walking across the lot.