To find the length of the building's shadow, we can use trigonometric functions in the context of the right triangle formed by the building, its shadow, and the line from the tip of the building to the tip of its shadow.
Given:
- Hypotenuse = 150 feet
- Angle at the top of the triangle = 75°
We'll denote:
- The length of the shadow as \( x \)
- The height of the building as \( h \)
Using the cosine function, which relates the angle, the length of the adjacent side (the shadow), and the hypotenuse:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
So:
\[ \cos(75^\circ) = \frac{x}{150} \]
Now, we can solve for \( x \):
\[ x = 150 \cdot \cos(75^\circ) \]
Next, we calculate \( \cos(75^\circ) \):
\[ \cos(75^\circ) \approx 0.2588 \]
Now plug this value back into the equation for \( x \):
\[ x \approx 150 \cdot 0.2588 \approx 38.82 \text{ feet} \]
However, that's not one of the provided responses. Let's check using the correct trigonometric function. Instead, we can also use the sine function for the height:
\[ \sin(75^\circ) = \frac{h}{150} \]
We don't need the height but this is another way to confirm the triangle relations. Given the responses, let’s look for cosine again for shadow length with a correction, focusing back to our trig identities correctly applied.
Let’s use sine since it's a shadow as noted directly in context:
Using \(\sin\):
\[ \sin(75^\circ) = \frac{\text{shadow}}{\text{hypotenuse}} \rightarrow \text{shadow} = 150 \cdot \sin(75^\circ) \] Using radians is an error context but for the expectation \( \sin(75^\circ) = 0.9659 \) gives us:
So then,
\[ x \approx 150 \cdot 0.9659 \approx 144.885 \text{ feet} \]
And since the closest option is:
144.9 feet.
The length of the building’s shadow is 144.9 feet.