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 100 paces long and 70 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 \boxed{}. the distance saved by using the diagonal path, up to three decimal places, is approximately \boxed{}

b. the distance saved by using the diagonal path is exactly \boxed{}. no approximation is necessary

Explanation:

Step1: Calculate the length of the two - side path

The length of the two - side path (walking along the length and the width) is \(l + w\), where \(l = 100\) paces and \(w=70\) paces. So, \(100 + 70=170\) paces.

Step2: Calculate the length of the diagonal path

Using the Pythagorean theorem \(d=\sqrt{l^{2}+w^{2}}\), where \(l = 100\) and \(w = 70\). Then \(d=\sqrt{100^{2}+70^{2}}=\sqrt{10000 + 4900}=\sqrt{14900}=10\sqrt{149}\) paces.

Step3: Calculate the distance saved

The distance saved \(s=(l + w)-d\). Substitute \(l = 100\), \(w = 70\) and \(d = 10\sqrt{149}\) into the formula. \(s=170-10\sqrt{149}\). Calculate \(10\sqrt{149}\approx10\times12.2066 = 122.066\). Then \(s=170-122.066 = 47.934\) paces.

Answer:

A. Using radicals, the distance saved by using the diagonal path is exactly \(170 - 10\sqrt{149}\). The distance saved by using the diagonal path, up to three decimal places, is approximately \(47.934\)