Question

a 50-foot tree casts a shadow 9 feet long. the sine of the angle between the ground and the line that connects the tip of the shadow to the top of the tree is approximately _____.

a. .02
b. .18
c. .98
d. .01

Explanation:

Step1: Identify the triangle type

The tree, its shadow, and the line from the tip of the shadow to the top of the tree form a right triangle. The tree height (50 feet) is the opposite side, the shadow length (9 feet) is the adjacent side, and the line is the hypotenuse.

Step2: Calculate the hypotenuse

Using the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \), where \( a = 50 \) and \( b = 9 \). So \( c=\sqrt{50^2 + 9^2}=\sqrt{2500 + 81}=\sqrt{2581}\approx50.80 \).

Step3: Calculate the sine of the angle

Sine of an angle in a right triangle is \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, opposite is 50, hypotenuse is approximately 50.80. So \( \sin\theta=\frac{50}{50.80}\approx0.98 \).

Answer:

C. .98