Question

a 680 - ft rope anchors a hot - air balloon as shown in the figure. (a) express the angle θ as a function of the height h of the balloon. θ = (b) find the angle θ if the balloon is 470 ft high. (round your answer to one decimal place.) θ =

Explanation:

Step1: Identify trigonometric relation

In the right - triangle formed, $\sin\theta=\frac{h}{680}$.

Step2: Solve for $\theta$ in part (a)

We can rewrite the equation as $\theta=\sin^{- 1}(\frac{h}{680})$.

Step3: Substitute $h = 470$ in part (b)

$\theta=\sin^{-1}(\frac{470}{680})$.

Step4: Calculate the value of $\theta$

$\theta=\sin^{-1}(0.691176)\approx43.8^{\circ}$

Answer:

(a) $\theta=\sin^{-1}(\frac{h}{680})$
(b) $43.8$