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

The expression for the angle is;

`\theta=\sin ^(-1)((h)/(680))`

The value of the angle is;

`61.9^0`

Step-by-step explanation:

Given the figure in the attached image.

The length of the rope is = 680 ft

a)

Using trigonometry;

Recall that;

`\sin \theta=(opposite)/(hypothenuse)`

From the diagram;

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

Substituting we have;

`\sin \theta=(h)/(680)`

Taking the sine inverse of both sides we have;

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

Therefore, the expression for the angle is;

`\theta=\sin ^(-1)((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)((h)/(680)) \\ \theta=\sin ^(-1)((600)/(680)) \\ \theta=61.9^0 \end{gathered}`

Therefore, the value of the angle is;

`61.9^0`