Question

Federico enjoys catching pokemons in university campus. One day, while trying to catch charmander, hefound the best spot next to a perfectly circular pond. He was 41 feet from the bank and 78 feet from thepoint of tangency. Determine the radius of the pond using the given information. Round to the nearestinteger.

Answer 1

Let the radius of the circular pond be x.

The pictorial representation of the problem is given below:

Recall the Tangent to a Circle Theorem:

It follows that we have a right triangle as shown:

Hence, the right triangle has legs with lengths 78 and x, and a hypothenuse of length 41+x.

Use the Pythagorean Theorem:

`x^2+78^2=(41+x)^2`

Solve the resulting equation for the radius x:

`\begin{gathered} \text{Simplify both sides of the equation:} \\ x^2+6084=x^2+82x+1681 \\ \text{Subtract }x^2\text{ from both sides:} \\ x^2+6084-x^2=x^2+82x+1681-x^2 \\ \Rightarrow6084=82x+1681 \\ \text{Swap the sides of the equation:} \\ \Rightarrow82x+1681=6084 \\ \text{Subtract 1681 from both sides:} \\ \Rightarrow82x+1681-1681=6084-1681 \\ \Rightarrow82x=4403 \\ \text{Divide both sides by 82:} \\ \Rightarrow(82x)/(82)=(4403)/(82) \\ \Rightarrow x=(4403)/(82)\approx54\text{ ft.} \end{gathered}`

Hence, the required radius of the pond is about 54 ft. to the nearest integer.