Question

A 680-ft rope anchors a hot-air balloon as shown in the figure. (a) Express the angle O as a function of the height h of the balloon. 0 = (b) Find the angle if the balloon is 600 ft high. (Round your answer to one decimal place.)

Answer 1

Answer:

The expression for the angle is;

$\theta=\sin ^{-1}(\frac{h}{680})$

The value of the angle is;

$61.9^0$

Explanation:

Given the figure in the attached image.

The length of the rope is = 680 ft

a)

Using trigonometry;

Recall that;

$\sin \theta=\frac{opposite}{hypothenuse}$

From the diagram;

$\begin{gathered} \text{Opposite = h} \\ \text{hypothenuse = 680 ft} \end{gathered}$

Substituting we have;

$\sin \theta=\frac{h}{680}$

Taking the sine inverse of both sides we have;

$\begin{gathered} \sin ^{-1}(\sin \theta)=\sin ^{-1}(\frac{h}{680}) \\ \theta=\sin ^{-1}(\frac{h}{680}) \end{gathered}$

Therefore, the expression for the angle is;

$\theta=\sin ^{-1}(\frac{h}{680})$

b)

Given that;

The balloon is 600 ft high

$h=600ft$

Substituting the value of h into the expression derived in question a;

$\begin{gathered} \theta=\sin ^{-1}(\frac{h}{680}) \\ \theta=\sin ^{-1}(\frac{600}{680}) \\ \theta=61.9^0 \end{gathered}$

Therefore, the value of the angle is;

$61.9^0$