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

$x^2 + (x + 16)^2 = 38^2$
Expand: $x^2 + x^2 + 32x + 256 = 1444$
Simplify: $2x^2 + 32x - 1188 = 0$ → $x^2 + 16x - 594 = 0$

Step2: Solve quadratic equation

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$a=1, b=16, c=-594$
Discriminant: $16^2 - 4(1)(-594) = 256 + 2376 = 2632$
$\sqrt{2632} \approx 51.303$
$x = \frac{-16 + 51.303}{2} \approx 17.651$ (positive root)

Step3: Calculate saved distance

Original distance: $x + (x + 16) = 2x + 16$
$2(17.651) + 16 \approx 51.302$
Saved distance: $51.302 - 38 \approx 13.302$

Step4: Round to nearest integer

$13.302 \approx 13$

Answer:

13