A straight road slopes at an angle of 10 degrees with the horizontal. When the angle of elevation of the sun (from horizontal) is 62.5 degrees, a telephone pole at the side of the road casts a 15 foot shadow downhill, parallel to the road. How high is the telephone pole?
A straight road slopes at an angle of 10° with the horizontal.When the angle of elevation of the sun is 62.5°,a telephone pole at the side of the road casts a 15-foot shadow downhill,parallel to the road.How high is the telephone pole?
16.7 ft
To find the height of the telephone pole, you can use trigonometry and the given information. Let's break down the problem step by step:
1. Draw a diagram: Sketch a right triangle to represent the situation. Label the horizontal base of the triangle as the length of the shadow (15 feet) and label the vertical side as the height of the pole (which we want to find).
2. Identify the known angles: One of the angles in the triangle is given as 10 degrees, which represents the slope of the road. The other known angle is the angle of elevation of the sun, which is given as 62.5 degrees.
3. Find the third angle of the triangle: To find the third angle, subtract the sum of the two known angles (10 + 62.5 = 72.5) from 180 degrees. This gives you the third angle, which is 180 - 72.5 = 107.5 degrees.
4. Use trigonometric ratios: In this case, we can use the tangent function since we know the angle and the length of the opposite and adjacent sides.
tan(θ) = height of pole / length of shadow
For the given triangle, tan(107.5 degrees) = height of pole / 15 feet.
5. Solve the equation for the height of the pole: Rearranging the equation, we have:
height of pole = tan(107.5 degrees) * 15 feet.
Using a calculator, find the tangent of 107.5 degrees and then multiply it by 15 feet to calculate the height of the pole.
6. Perform the calculation: Using a calculator, tan(107.5 degrees) * 15 feet ≈ 85.84 feet.
Therefore, the height of the telephone pole is approximately 85.84 feet.