To find the horizontal distance of the ramp, we can use the trigonometric relationship involving the sine function, which relates the height of the ramp to the angle of elevation and the hypotenuse.
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We know that:
- The height (opposite side) is 2.5 feet.
- The angle of elevation (θ) is 14°.
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We need to find the length of the horizontal distance (adjacent side). We can use the tangent function, which is defined as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Rearranging this gives us the formula for the adjacent side: \[ \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \]
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Substitute the known values into the formula: \[ \text{adjacent} = \frac{2.5}{\tan(14^\circ)} \]
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Calculate \( \tan(14^\circ) \) using a calculator: \[ \tan(14^\circ) \approx 0.2493 \]
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Now, calculate the horizontal distance: \[ \text{adjacent} \approx \frac{2.5}{0.2493} \approx 10.03 \]
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Round to the nearest tenth: \[ \text{adjacent} \approx 10.0 , \text{feet} \]
Thus, the horizontal distance of the ramp is approximately 10.0 feet.