Question

Your friend is flying a hot air balloon and you are recording them from below. When you start filming, the hot air balloon is at an angle of 63 degrees above you while staying 100 feet away from where it launched. After 5 seconds, the angle has increased to 67.9 degrees. How fast is your friend rising in feet per seconds and miles per hour.

Answer 1

10 ft/s and 6.818 mi/hr

Step-by-step explanation:

We can represent the situation with the following diagram:

When the angle is 63 degrees, the value of h can be calculated using a trigonometric function as:

`\tan 63=(h_i)/(100)`

Because h is the opposite side and 100 ft is the adjacent side. Solving for h, we get:

`\begin{gathered} h_i=100*\tan 63 \\ h_i=196.26\text{ ft} \end{gathered}`

In the same way, when the angle is 67.9 degrees, we can calculate the height as follows:

`\begin{gathered} \tan 67.9=(h_f)/(100) \\ h_f=100*\tan 67.9 \\ h_f=246.27ft \end{gathered}`

Now, we can calculate the speed in ft per second as follows:

`s=(h_f-h_i)/(t)=(246.27ft-196.24ft)/(5s)=10\text{ ft/s}`

Finally, 1 mile = 5280 ft and 1 hour = 3600 seconds, so we can convert to miles per hour as:

`10\text{ ft/s }*\frac{1\text{ mile}}{5280\text{ ft}}*\frac{3600}{1\text{ hour}}=6.818\text{ mi/hr}`

Therefore, the answers are:

10 ft/s and 6.818 mi/hr