Math Problem Help

Question

Marsha and Edwin want to know how far they will have to swim to get across a small pond. They have a 100-foot rope that they place at the waters edge (in a straight line), then standing at each end, they use a compass to measure the angle between the rope and a tree on the other side of the lake. Marsha’s angle is 67°. Edwin’s angle is 81°. How far will they have to swim to get to the other side of the pond?

Answer 1

Note: This is a nice problem that requires trigonometry and algebra to solve. The solution requires a slight assumption that the question is asking for the shortest distance they need to swim as otherwise there could be many answers.

Visualization and Notation

Visualize the straight line formed by the rope on one side of the pond and the tree as a dot on the other side of the pond that Marsha and Edwin want to swim to.

The shortest distance to swim would be the perpendicular line you can drop from the dot to the straight line made by the rope. Let that distance be = h.

Let's also denote the point where the perpendicular meets the rope to be point C with point A as point where Marsha is measuring and point B the point where Edwin is measuring.

Then,

tan(67°) = h/CA ⇒ CA = h * cot(67°)

tan(81°) = h/CB ⇒ CB = h * cot(81°)

We know that CA + CB = 100 feet

Therefore,

100 = CA + CB

100 = h * [ cot(67°) + cot(81°)]

or h = 100 / [ cot(67°) + cot(81°)] ... answer

Putting in values...

h = 100 / (0.424 + 0.158)

= 172 feet approximately