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Question Video: Using Right Triangle Trigonometry to Find Lengths in Word Problems Mathematics • 11th Grade

A man who is 1.7 meters tall is standing in front of a 4.3-meter-high streetlight. When the streetlight is turned on, the man’s shadow is 2.2 meters long. Find the distance between the man and the base of the streetlight, giving the answer to two decimal places.

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Video Transcript

A man who is 1.7 meters tall is standing in front of a 4.3-meter-high streetlight. When the streetlight is turned on, the man’s shadow is 2.2 meters long. Find the distance between the man and the base of the streetlight, giving the answer to two decimal places.

Sketching what we already know, the height of our streetlight and the height of the man, we can then add a line to represent the shadow when the street lamp turns on of length 2.2 meters. Our unknown value is how far the man is from the base of the lamppost. Let’s call this 𝑥 meters.

We can think about what’s happening here in terms of right triangles. Even though our image isn’t drawn perfectly to scale, we can see that the large right triangle could have a side length of 4.3 to represent the height of the streetlight. And there exists a smaller right triangle inside the larger one such that it has a side length of 1.7 representing the height of the man and an additional side of 2.2 meters representing the shadow. The unknown value, the distance between the man and the base of the street lamp, is 𝑥. Additionally, we can see that these triangles share an angle, which we’ll call angle 𝜃. This is a shared angle of elevation between the top of the streetlight and the height of the man.

To find the value of 𝑥, let’s consider some trig ratios. We know that sine equals the opposite over the hypotenuse, cosine equals the adjacent over the hypotenuse, and tangent equals the opposite over the hypotenuse. This means we can write a relationship for tan of 𝜃 equal to 1.7 over 2.2. If we consider the larger triangle that shares this angle 𝜃, we can write a relationship for the tan of 𝜃, where the adjacent side length is equal to 𝑥 plus 2.2. This gives us a second equation for tan of 𝜃 that is equal to 4.3 over 𝑥 plus 2.2.

Since we’re dealing with the same angle, 𝜃, we can set these two ratios equal to each other. We would have 1.7 over 2.2 is equal to 4.3 over 𝑥 plus 2.2. If we multiply through on both sides by 2.2 and then multiply through on both sides by 𝑥 plus 2.2, we’ll get 1.7 times 𝑥 plus 2.2 is equal to 2.2 times 4.3. Distributing our 1.7 and then multiplying 2.2 by 4.3 gives us 1.7𝑥 plus 3.74 equals 9.46. Subtracting 3.74 gives us 1.7𝑥 is equal to 5.74. Dividing through by 𝑥 gives us 𝑥 equals 3.3647 continuing.

We wanna round to two decimal places here. When we do that, we get 3.36. Remember, the context of our question is how far the man is from the base of the streetlight. And that’s going to be 3.36 meters.

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