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$\newline$ You are skiing on a mountain with an altitude of $1200$ meters. The angle of depression is $21^{\circ}$ . How far do you ski down the mountain? (Round to the nearest meter) $\newline$ Select the correct response: $\newline$ $1873$ meters $\newline$ $3349$ meters $\newline$ $1285$ meters

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Understanding integers

Full solution

Q.

\newline

You are skiing on a mountain with an altitude of

1200

meters. The angle of depression is

21^{\circ}

. How far do you ski down the mountain? (Round to the nearest meter)

\newline

Select the correct response:

\newline

1873

meters

\newline

3349

meters

\newline

1285

meters

Understand and Visualize: Understand the problem and visualize the scenario. $\newline$ We have a right triangle where the altitude of the mountain forms one side (the opposite side to the angle of depression), and we need to find the length of the slope (the hypotenuse) that the skier will ski down. The angle of depression from the horizontal is given as $21$ degrees, which is also the angle of elevation from the base to the top of the mountain.
Use Trigonometry: Use trigonometry to solve for the hypotenuse. $\newline$ We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle. The formula is: $\newline$ $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$ $\newline$ Here, the angle is $21$ degrees, and the opposite side is the altitude of the mountain, which is $1200$ meters. We need to find the adjacent side, which is the distance down the mountain.
Rearrange Formula: Rearrange the formula to solve for the adjacent side (distance down the mountain). $\newline$ $\text{adjacent} = \frac{\text{opposite}}{\tan(\text{angle})}$ $\newline$ $\text{adjacent} = \frac{1200 \, \text{meters}}{\tan(21 \, \text{degrees})}$
Calculate Distance: Calculate the distance using the tangent of $21$ degrees. $\newline$ First, we need to find the value of $\tan(21^\circ)$ . Using a calculator, we find: $\newline$ $\tan(21^\circ) \approx 0.383864$ $\newline$ Now, we can calculate the adjacent side: $\newline$ $\text{adjacent} = \frac{1200 \text{ meters}}{0.383864} \approx 3127.65 \text{ meters}$

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