a We know that the tallest flagpole in the world was in San Francisco. When the angle of elevation of the sun was 55^(∘), the length of the shadow cast by this flagpole was 210 feet.
Notice that we have a right triangle. Then, we can use the trigonometric ratio for tangent to find the height of the flagpole.
tan θ = Opposite/Adjacent
In our case, the measure of θ is 55^(∘), the length of the adjacent leg is 210, and the length of the opposite leg is x.
tan θ = Opposite/Adjacent ⇒ tan 55^(∘) = x/210
Let's solve this equation to find the length of opposite leg!
Therefore, the height of the flagpole is about 300 feet.
b We want to find the length of the shadow when the angle of elevation of the sun is 34^(∘). To do so, we will use again the trigonometric ratio for tangent.
tan θ = x/210
This time we will substitute θ= 34^(∘) into this equation and solve it for x. Let's do it!
Therefore, the height of the flagpole will be approximately 142 feet.
c To solve the problems we need to use the trigonometric ratios. But to use them, we need to assume that we have a right triangle formed by the shadow and the flagpole. This is only possible if the flagpole and the shadows are perpendicular to each other.
Flagpole ⊥ Shadow
Only with this assumption, we will be able to use trigonometric ratios.
