Question

Trey wants to use string lights to build a flagpole lighted tree in his front yard. Trey already had a 7 ft tall pole mounted in the ground. If the angle each light string makes with the ground is 73.5°, use the sine ratio to determine how long each light string must be to reach the top of the pole. Round your answer to the nearest tenth.(1 point) Responses 6.7 feet 6.7 feet 23.6 feet 23.6 feet 2 feet 2 feet 7.3 feet

Answer 1

To determine the length of each light string, we can use the sine ratio from trigonometry. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.

In this case:
- The height of the pole (the opposite side) is 7 feet.
- Let's denote the length of the light string (the hypotenuse) as \( L \).
- The angle with the ground is \( 73.5^\circ \).

The sine function can be expressed as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Substituting the known values:
\[
\sin(73.5^\circ) = \frac{7}{L}
\]

To find \( L \):
\[
L = \frac{7}{\sin(73.5^\circ)}
\]

Now, we calculate \( \sin(73.5^\circ) \):

Using a calculator:
\[
\sin(73.5^\circ) \approx 0.9563
\]

Now plug this value into the equation for \( L \):
\[
L = \frac{7}{0.9563} \approx 7.33
\]

Rounding to the nearest tenth gives:
\[
L \approx 7.3 \text{ feet}
\]

Therefore, the length of each light string must be approximately **7.3 feet**.