To find the length of each light string needed to reach the top of the pole using the sine ratio, we can set up the following relationship based on the right triangle formed by the pole and the light string.
Let:
- \( h = 7 \) feet (height of the pole)
- \( L \) = length of the light string
- \( \theta = 73.5^\circ \) (angle the light string makes with the ground)
According to the sine function: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] In this scenario:
- The "opposite" side is the height of the pole \( h \).
- The "hypotenuse" is the length of the light string \( L \).
We can rearrange the sine ratio to solve for \( L \): \[ L = \frac{h}{\sin(\theta)} \]
Now, substitute the values: \[ L = \frac{7}{\sin(73.5^\circ)} \]
Next, calculate \( \sin(73.5^\circ) \): Using a calculator, \[ \sin(73.5^\circ) \approx 0.9563 \]
Now plug this into the equation: \[ L \approx \frac{7}{0.9563} \approx 7.32 \]
Rounding to the nearest tenth, we find: \[ L \approx 7.3 \text{ feet} \]
Thus, the length of each light string must be approximately 7.3 feet.
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