Question

use the pythagorean theorem and the square - root property to solve the following problem. express your answer in simplified radical form. then find a decimal approximation to the nearest tenth.
a rectangular park is 12 miles long and 4 miles wide. how long is a pedestrian route that runs diagonally across the park?
in simplified radical form, the pedestrian route is (square) miles long.

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem for a right - triangle is \(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the legs of the right - triangle and \(c\) is the hypotenuse. In a rectangle, the length and width form the legs of a right - triangle, and the diagonal is the hypotenuse. Here \(a = 12\) miles and \(b = 4\) miles.

Step2: Substitute values into the formula

We have \(12^{2}+4^{2}=c^{2}\). Calculate \(12^{2}=144\) and \(4^{2}=16\). Then \(144 + 16=c^{2}\), so \(c^{2}=160\).

Step3: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{160}\). Simplify \(\sqrt{160}=\sqrt{16\times10}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 16\), \(b = 10\)), we get \(c = 4\sqrt{10}\) miles.

Step4: Find decimal approximation

\(\sqrt{10}\approx3.162\), so \(4\sqrt{10}\approx4\times3.162 = 12.648\approx12.6\) miles.

Answer:

Simplified radical form: \(4\sqrt{10}\) miles. Decimal approximation: \(12.6\) miles.