Question

Jamal is hanging up Christmas lights in his front yard to mimic the look of a Christmas tree. He starts by placing a pole in the ground and then extending strands of lights from the top of the pole to the ground in a circular pattern. The pole is 9 feet tall once placed in the ground, and each light strand is 16 feet long. If he attaches a strand of lights to the top of the pole and extends it down to the ground, what is the angle formed between the light strand and the pole? Round your answer to the nearest tenth.

Answer 1

To find the angle formed between the light strand and the pole, we can use some basic trigonometry.

Let's denote:
- the height of the pole as \( h = 9 \) feet,
- the length of the light strand as \( L = 16 \) feet.

When Jamal attaches the strand of lights to the top of the pole and extends it to the ground, we can visualize this situation as a right triangle. The height of the pole (\( 9 \) feet) is one leg of the triangle, the length of the light strand (\( 16 \) feet) is the hypotenuse, and the horizontal distance from the base of the pole to the point where the light strand touches the ground is the other leg.

We will use the sine function to find the angle \( \theta \) between the pole and the light strand. The sine of the angle is given by the ratio of the opposite side (the height of the pole) to the hypotenuse (the length of the light strand):

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L} = \frac{9}{16}
\]

Now, we will find \( \theta \) by taking the inverse sine:

\[
\theta = \arcsin\left(\frac{9}{16}\right)
\]

Using a calculator, we compute:

\[
\theta \approx \arcsin(0.5625)
\]

Calculating the value gives:

\[
\theta \approx 34.9^\circ
\]

Thus, the angle formed between the light strand and the pole is approximately

\[
\boxed{34.9}
\] degrees, rounded to the nearest tenth.