Question

1. a rope from the top of a mast on a sailboat is attached to a point 19 feet from the mast. if the rope is 28 feet long, how tall is the mast? round to the nearest tenth of a foot.

Explanation:

Step1: Apply Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the length of the hypotenuse (the rope length, \(c = 28\) feet) and \(a\) is the distance from the mast (\(a=19\) feet), and \(b\) is the height of the mast. We need to solve for \(b\). Rearranging the formula gives \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute values

Substitute \(c = 28\) and \(a = 19\) into the formula: \(b=\sqrt{28^{2}-19^{2}}=\sqrt{(28 + 19)(28-19)}\) (using \(x^{2}-y^{2}=(x + y)(x - y)\)). Then \(b=\sqrt{47\times9}=\sqrt{423}\).

Step3: Calculate the square - root

\(\sqrt{423}\approx20.6\)

Answer:

The height of the mast is approximately \(20.6\) feet.