A 25-foot-long ladder leans against a vertical wall, forming an angle of with the level ground. To the nearest tenth of a foot, how high above the ground does the ladder touch the wall?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 35 | Free SAT Prep | Aniko

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Question 35·Medium·Right Triangles and Trigonometry

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A 25-foot-long ladder leans against a vertical wall, forming an angle of $65^{\circ}$ with the level ground. To the nearest tenth of a foot, how high above the ground does the ladder touch the wall?

For right-triangle word problems like ladders against walls, first sketch and label the triangle so you clearly see which side is opposite, adjacent, and the hypotenuse relative to the given angle. Then choose the trig ratio (sine: opposite/hypotenuse, cosine: adjacent/hypotenuse, tangent: opposite/adjacent) that involves the known side and the side you are solving for. Set up a simple equation, solve for the unknown, and use your calculator in degree mode, rounding only at the final step to match the SAT’s requested precision.

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Identify which side you are solving for

Think about the triangle formed by the wall, the ground, and the ladder. Is the height on the wall opposite, adjacent to, or the hypotenuse relative to the $65^\circ$ angle?

Pick the correct trig function

You know the hypotenuse (25 ft) and you want the side opposite the $65^\circ$ angle. Which trig ratio uses opposite and hypotenuse: sine, cosine, or tangent?

Set up and solve the equation

Write an equation using your chosen trig function equal to $\dfrac{\text{unknown height}}{25}$ . Then solve the equation and remember to have your calculator in degree mode and to round to the nearest tenth.

Desmos Guide

Enter the trigonometric expression

In Desmos, type 25*sin(65) (make sure the calculator is set to degrees, not radians).

Use the output and round appropriately

Look at the numerical result that Desmos gives for 25*sin(65), then round that value to the nearest tenth to match one of the answer choices.

Step-by-step Explanation

Visualize and label the right triangle

Draw a right triangle to represent the situation:

The ground is the horizontal leg.
The wall is the vertical leg.
The ladder is the hypotenuse, with length 25 ft.
The angle between the ladder and the ground is $65^\circ$ .

We are asked for the height where the ladder touches the wall, which is the vertical leg (the side opposite the $65^\circ$ angle).

Choose the correct trigonometric ratio

In a right triangle:

Opposite the angle: the vertical height up the wall (what we want).
Hypotenuse: the ladder, 25 ft.

The trig function that relates opposite and hypotenuse is sine:

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

So we will use $\sin(65^\circ)$ to find the height.

Set up the sine equation

Let $h$ be the height (in feet) where the ladder touches the wall. Using the sine ratio:

\sin(65^\circ) = \frac{h}{25}

Now solve for $h$ by multiplying both sides by 25:

h = 25 \cdot \sin(65^\circ)

Calculate and round to the nearest tenth

Use a calculator in degree mode to evaluate $25 \cdot \sin(65^\circ)$ .

First find $\sin(65^\circ)$ (it should be a little more than $0.9$ ).
Then multiply that result by 25 to get $h$ .
You should get a value close to $22.66$ feet.

Rounding this to the nearest tenth gives $22.7$ feet, which matches choice D) 22.7.

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