Question

the varners live on a corner lot. often, children cut across their lot to save walking distance. the diagram to the right represents the corner lot. the childrens path is represented by a dashed line. approximate the walking distance that is saved by cutting across their property instead of walking around the lot. the walking distance that is saved by cutting across the lot is □ (round the final answer to the nearest integer as needed. round all intermediate values to the nearest thousandth as needed)

Explanation:

Step1: Apply Pythagorean theorem

By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), for right - triangle with legs \(x\) and \(x + 10\) and hypotenuse \(48\). So \(x^{2}+(x + 10)^{2}=48^{2}\).
Expand: \(x^{2}+x^{2}+20x + 100=2304\).
Combine like terms: \(2x^{2}+20x+100 - 2304=0\), \(2x^{2}+20x - 2204 = 0\), divide by \(2\): \(x^{2}+10x-1102=0\).
Use quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) with \(a = 1\), \(b = 10\), \(c=-1102\).
\(x=\frac{-10\pm\sqrt{100+4408}}{2}=\frac{-10\pm\sqrt{4508}}{2}=\frac{-10\pm67.157}{2}\).
Take positive root \(x=\frac{-10 + 67.157}{2}=\frac{57.157}{2}=28.579\).

Step2: Calculate distances

Distance around the lot: \(x+(x + 10)=2x + 10\). Substitute \(x = 28.579\), \(2\times28.579+10=57.158+10=67.158\).
Distance across the lot: \(48\).
Distance saved: \(67.158-48=19.158\approx18\) (rounded to nearest integer).

Answer:

\(18\) feet