To find the angle formed between the top of the building and the shadow, we can use trigonometry. The scenario involves a right triangle where:
- The opposite side (height of the building) is unknown.
- The adjacent side (length of the shadow) is 13 feet.
- The hypotenuse (diagonal distance) is 15 feet.
To find the angle, we need to determine the height of the building using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \] Where:
- \(c\) is the hypotenuse (15 feet),
- \(a\) is the height of the building,
- \(b\) is the shadow (13 feet).
Let's calculate the height (a):
\[ 15^2 = a^2 + 13^2 \] \[ 225 = a^2 + 169 \] \[ a^2 = 225 - 169 \] \[ a^2 = 56 \] \[ a = \sqrt{56} \approx 7.48 \text{ feet} \]
Now we can find the angle using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{7.48}{15} \]
Calculating the sine:
\[ \sin(\theta) \approx \frac{7.48}{15} \approx 0.4987 \]
Now using the inverse sine function:
\[ \theta \approx \sin^{-1}(0.4987) \approx 29.9^\circ \approx 30^\circ \]
Therefore, the approximate angle formed between the top of the building and the shadow is 30°.
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