Question

1. AI = 400 ft2. Calculate the distance MI for the length of the zipline cable. 3. Calculate the angle at which our zipliners will be descending toward the island . Safety regulations state that the angle at which a zipline cable meets the launching point cannot be smaller than 68 degrees . Please determine if we are in compliance with these regulations

Answer 1

2. The Pythagorean theorem states:

`c^2=a^2+b^2`

where a and b are the legs and c is the hypotenuse of a right triangle.

Applying this theorem to triangle AMI (where AI and MA are the legs and MI is the hypotenuse), we get:

`\begin{gathered} MI^2=AI^2+MA^2 \\ MI^2=400^2+100^2 \\ MI^2=160000+10000 \\ MI^2=170000 \\ MI=\sqrt[]{170000} \\ MI\approx412.31\text{ ft} \end{gathered}`

3. By definition:

`\tan (angle)=\frac{\text{opposite}}{\text{adjacent}}`

Applying this definition to triangle AMI, considering the angle M, we get:

`\begin{gathered} \tan (\angle M)=(AI)/(MA) \\ \tan (\angle M)=(400)/(100) \\ \tan (\angle M)=4 \\ \angle M=\arctan (4) \\ \angle M\approx76\text{ \degree} \end{gathered}`

This angle is greater than 68°, then it satisfies the regulation.