SAT Right Triangles & Trigonometry Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Right Triangles and Trigonometry

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A surveyor stands 50 meters from the base of a communications tower and measures the angle of elevation to the top of the tower to be $35^{\circ}$ . Which of the following is closest to the height of the tower, in meters?

(Note: $\sin 35^{\circ}\approx 0.57$ , $\cos 35^{\circ}\approx 0.82$ , and $\tan 35^{\circ}\approx 0.70$ .)

For right-triangle angle-of-elevation problems, first sketch the triangle and clearly label the angle, the given side, and the side you need. Decide whether the unknown is opposite, adjacent, or hypotenuse relative to the given angle, and then choose the trig function (sine, cosine, or tangent) that directly relates the known side to the unknown. Write a simple equation (like $\tan(\theta)=\frac{\text{opp}}{\text{adj}}$ ), solve for the unknown in one step, and only then use any provided trig approximations or your calculator to compute and match to the closest choice. This avoids common mistakes like using the wrong trig ratio or solving for the hypotenuse instead of the vertical height.

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Identify the sides of the triangle

Draw a right triangle with the surveyor, the base of the tower, and the top of the tower. Which side is the 50 meters, and which side represents the height of the tower?

Pick the right trig function

You know the side next to the angle $35^\circ$ (the 50 m) and you want the side across from that angle (the height). Which trig function compares the opposite side to the adjacent side?

Use the given trig values

Once you have an equation involving $\tan(35^\circ)$ , use the approximation $\tan 35^\circ \approx 0.70$ to solve for the unknown height.

Desmos Guide

Check the tangent calculation

In Desmos, type the expression 50 * tan(35°) (or 50 * 0.70 using the given approximation). Look at the numerical output and then choose the answer choice that is closest to this value.

Step-by-step Explanation

Draw and label the right triangle

Imagine a right triangle where:

One vertex is at the surveyor.
One vertex is at the base of the tower.
The third vertex is at the top of the tower.

The surveyor is 50 meters from the base, so the horizontal side from surveyor to base is 50 m. The angle at the surveyor between the ground and the line of sight to the top is $35^\circ$ . The height of the tower is the vertical side opposite this $35^\circ$ angle.

Choose the correct trigonometric ratio

Relative to the $35^\circ$ angle at the surveyor:

The opposite side is the tower’s height (what we want).
The adjacent side is the 50 m horizontal distance (what we know).

The trig ratio that uses opposite and adjacent is tangent:

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

So for this triangle:

\tan(35^\circ) = \frac{\text{height}}{50}

Solve the equation for the height

Start with the equation:

\tan(35^\circ) = \frac{\text{height}}{50}

Multiply both sides by 50 to isolate the height:

\text{height} = 50\tan(35^\circ)

Use the approximation from the problem, $\tan 35^\circ \approx 0.70$ :

\text{height} \approx 50 \times 0.70

\text{height} \approx 35

Match the result to the answer choices

The calculation shows the tower’s height is about 35 meters. Among the choices 29, 35, 41, and 61, the value that matches this result is 35, so that is the correct answer.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation