Question

1. from a hot - air balloon, isaiah measures a 22° angle of depression to a landmark thats 312 feet away, measuring horizontally. whats the balloons vertical distance above the ground? round your answer to the nearest hundredth of a foot if necessary.
2. a boat is heading towards a lighthouse, where benjamin is watching from a vertical angle of 113 feet above the water. benjamin measures an angle of depression to the boat at point a to be 17°. at some later time, benjamin takes another measurement and finds the angle of depression to the boat (now at point b) to be 70°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Set up tangent equation for problem 14

We know that the angle of depression is \(22^{\circ}\) and the horizontal distance is \(312\) feet. Let the vertical distance (height of the balloon above the ground) be \(h\). The angle of depression is equal to the angle of elevation from the landmark to the balloon. Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), we have \(\tan(22^{\circ})=\frac{312}{h}\).

Step2: Solve for \(h\)

We can re - arrange the equation \(\tan(22^{\circ})=\frac{312}{h}\) to \(h = \frac{312}{\tan(22^{\circ})}\). Since \(\tan(22^{\circ})\approx0.404\), then \(h=\frac{312}{0.404}\approx772.28\) feet.

Step3: Set up tangent equations for problem 15

Let the distance from the base of the lighthouse to point \(A\) be \(d_1\) and to point \(B\) be \(d_2\). For the angle of depression of \(17^{\circ}\) from the lighthouse to point \(A\), \(\tan(17^{\circ})=\frac{113}{d_1}\), so \(d_1=\frac{113}{\tan(17^{\circ})}\). Since \(\tan(17^{\circ})\approx0.3057\), \(d_1=\frac{113}{0.3057}\approx370\) feet. For the angle of depression of \(70^{\circ}\) from the lighthouse to point \(B\), \(\tan(70^{\circ})=\frac{113}{d_2}\), so \(d_2 = \frac{113}{\tan(70^{\circ})}\). Since \(\tan(70^{\circ})\approx2.7475\), \(d_2=\frac{113}{2.7475}\approx41\) feet.

Step4: Find the distance between \(A\) and \(B\)

The distance between \(A\) and \(B\) is \(d = d_1 - d_2\). So \(d=370 - 41=329\) feet.

Answer:

1. The balloon's vertical distance above the ground is approximately \(772.28\) feet.
2. The distance from point \(A\) to point \(B\) is approximately \(329\) feet.