Question

question 3
1 pts
an airplane is flying at an altitude of 15,000 ft.
the angle of depression to a point on the ground is 21°.
how far must the plane travel to be directly above the point?
round your answer to the nearest foot.

question 4
1 pts
a surveyors instrument is 5 ft above the ground.
the distance to a tree is 100 ft, and the angle of elevation to the
top is 48°.
how tall is the tree?
round your answer to the nearest hundredth.

Explanation:

Question 3

Step1: Use the tangent function

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The angle of depression is \(21^{\circ}\), and the opposite side (altitude) is \(15000\) ft. Let \(x\) be the horizontal distance (the distance the plane must travel). Since the angle of depression is equal to the angle of elevation from the point on the ground, \(\tan(21^{\circ})=\frac{15000}{x}\).

Step2: Solve for \(x\)

$$x = \frac{15000}{\tan(21^{\circ})}$$

Using a calculator, \(\tan(21^{\circ})\approx0.3839\), so \(x=\frac{15000}{0.3839}\approx39073\) ft.

Question 4

Step1: Use the tangent function for the height above the instrument

Let \(h\) be the height above the \(5\) - ft instrument. We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 48^{\circ}\) and the adjacent side is \(100\) ft. So \(\tan(48^{\circ})=\frac{h}{100}\).

Step2: Solve for \(h\)

$$h = 100\times\tan(48^{\circ})$$

Using a calculator, \(\tan(48^{\circ})\approx1.1106\), so \(h = 100\times1.1106=111.06\) ft.

Step3: Find the total height of the tree

The total height of the tree \(H=h + 5\). So \(H=111.06+5=116.06\) ft.

Answer:

Question 3: \(39073\) ft
Question 4: \(116.06\) ft