A building casts a shadow reaching 13 feet from the base of the building, with a diagonal distance of 15 feet from the top of the building. Using the inverse of sine, what is the approximate angle formed between the top of the building and the shadow?(1 point)

Question

Responses

45°
45°

90°
90°

60°
60°

30°
30°

Answer 1

To find the angle formed between the top of the building and the shadow, we can use the definition of sine in a right triangle.

In this case, we have:

• The opposite side (the height of the building, which we will denote as \(h\)).
• The adjacent side (the shadow length, which is 13 feet).
• The hypotenuse, which is the diagonal distance from the top of the building to the end of the shadow at the ground, measuring 15 feet.

Using the Pythagorean theorem to find the height of the building: \[ h^2 + 13^2 = 15^2 \] \[ h^2 + 169 = 225 \] \[ h^2 = 225 - 169 = 56 \] \[ h = \sqrt{56} \approx 7.48 \text{ feet} \]

Now, we can apply the sine function to find the angle \( \theta \): \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{15} \] \[ \sin(\theta) = \frac{7.48}{15} \approx 0.4987 \]

Now, we find \( \theta \) using the inverse sine function: \[ \theta = \sin^{-1}(0.4987) \approx 29.9^\circ \approx 30^\circ \]

Therefore, the angle formed between the top of the building and the shadow is approximately 30°. So the correct response is:

30°