Question

students at college have worn a path that cuts diagonally across the campus \quad\. if the quad is a rectangle that paisley measures as 130 paces long and 80 paces wide, how many paces will paisley save by using the diagonal path?
select the correct choice and fill in the answer box(es) to complete your choice.
a. using radicals, the distance saved by using the diagonal path is exactly (square). the distance saved by using the diagonal path, up to three decimal places, is approximately (square)
b. the distance saved by using the diagonal path is exactly (square). no approximation is necessary

Explanation:

Step1: Find the length of the diagonal

Use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 130\) and \(b=80\).
\(c=\sqrt{130^{2}+80^{2}}=\sqrt{16900 + 6400}=\sqrt{23300}\)

Step2: Calculate the distance saved

The non - diagonal path length is \(130 + 80=210\).
The distance saved \(d=(130 + 80)-\sqrt{130^{2}+80^{2}}=210-\sqrt{23300}\)
\(\sqrt{23300}=\sqrt{100\times233}=10\sqrt{233}\)
\(d = 210-10\sqrt{233}\) (exact value, no approximation needed as per option B)

Answer:

B. The distance saved by using the diagonal path is exactly \(\boldsymbol{210 - 10\sqrt{233}}\). No approximation is necessary.