Question

1. steps to the entrance of a school rise a total of 2 feet. they are to be torn out and replaced with a wheelchair ramp, inclined at 12°. how long is the surface of the ramp, to the nearest foot?
2. the angle of depression from the top of a cliff to a boat in the water below is 21°. the boat is anchored 208 meters from the base of the cliff. determine the height of the cliff, to the nearest tenth of a meter.
3. henry is going on a hike. he parks his car and starts his hike by going 8 kilometers directly south, then 5 kilometers west. he notices the sun going down, so he wants to get back as fast as he can before it gets dark! what is the angle between his hiked path back to his starting point? round to the nearest degree.

Explanation:

4.

Step1: Use the sine function

We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 12^{\circ}\), the opposite side (height) \(y = 2\) feet, and the hypotenuse is the length of the ramp \(x\). So, \(\sin(12^{\circ})=\frac{2}{x}\).

Step2: Solve for \(x\)

$$x=\frac{2}{\sin(12^{\circ})}$$

Since \(\sin(12^{\circ})\approx0.2079\), then \(x=\frac{2}{0.2079}\approx9.6\approx10\) feet.

5.

Step1: Use the tangent function

The angle of depression is \(21^{\circ}\), and the adjacent side (distance from the base of the cliff) \(x = 208\) meters. Let the height of the cliff be \(y\). The angle of depression is equal to the angle of elevation from the boat to the top of the cliff. Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(21^{\circ})=\frac{y}{208}\).

Step2: Solve for \(y\)

$$y = 208\times\tan(21^{\circ})$$

Since \(\tan(21^{\circ})\approx0.3839\), then \(y=208\times0.3839 = 80.0512\approx80.1\) meters.

6.

Step1: Use the tangent function

Henry's path forms a right - triangle. The vertical side (south direction) \(a = 8\) km and the horizontal side (west direction) \(b = 5\) km. Let the angle between his hiked path back (hypotenuse) and the south - ward path be \(\theta\). Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), here \(\tan\theta=\frac{5}{8}\).

Step2: Find \(\theta\)

\(\theta=\arctan(\frac{5}{8})\)
Since \(\frac{5}{8}=0.625\), \(\theta=\arctan(0.625)\approx32^{\circ}\)

Answer:

1. The length of the ramp is \(10\) feet.
2. The height of the cliff is \(80.1\) meters.
3. The angle is \(32^{\circ}\).