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.
