Question

6.) you are standing 50 meters from a hot air balloon that is preparing to take off. the angle of elevation to the top of the balloon is 28°. find the height of the balloon.
8.) john wants to measure the height of a tree. he walks exactly 100 feet from the base of the tree and looks up. the angle from the ground to the top of the tree is 33°. how tall is the tree?
10.) kelly is flying a kite to which the angle of elevation is 70°. the string on the kite is 65 meters long. how far is the kite above the ground?

Explanation:

Step1: Set up tangent - ratio for balloon problem

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For the hot - air balloon problem, $\theta = 28^{\circ}$ and the adjacent side to the angle of elevation is $x = 50$ meters. Let the height of the balloon be $h_1$. Then $\tan(28^{\circ})=\frac{h_1}{50}$.

Step2: Solve for $h_1$

$h_1 = 50\times\tan(28^{\circ})$. Using a calculator, $\tan(28^{\circ})\approx0.5317$, so $h_1=50\times0.5317 = 26.585\approx26.6$ meters.

Step3: Set up tangent - ratio for tree problem

For the tree problem, $\theta = 33^{\circ}$ and the adjacent side to the angle of elevation is $x = 100$ feet. Let the height of the tree be $h_2$. Then $\tan(33^{\circ})=\frac{h_2}{100}$.

Step4: Solve for $h_2$

$h_2 = 100\times\tan(33^{\circ})$. Using a calculator, $\tan(33^{\circ})\approx0.6494$, so $h_2 = 100\times0.6494=64.94\approx64.9$ feet.

Step5: Set up sine - ratio for kite problem

We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For the kite problem, $\theta = 70^{\circ}$ and the hypotenuse (length of the string) is $l = 65$ meters. Let the height of the kite above the ground be $h_3$. Then $\sin(70^{\circ})=\frac{h_3}{65}$.

Step6: Solve for $h_3$

$h_3=65\times\sin(70^{\circ})$. Using a calculator, $\sin(70^{\circ})\approx0.9397$, so $h_3 = 65\times0.9397=61.0805\approx61.1$ meters.

Answer:

• Height of the balloon: Approximately $26.6$ meters.
• Height of the tree: Approximately $64.9$ feet.
• Height of the kite above the ground: Approximately $61.1$ meters.