A straight road slopes upward 14 degrees from the horizontal. A vertical telephone pole beside the road casts a shadow of 60 feet down the road. if the angle of elevation of the sun is 55 degrees, what is the height of the telephone pole?
draw a diagram.
Label these points:
T = top of pole
B = base of pole on road
S = tip of shadow on road
Draw a horizontal line from S to where it intersects the extension of TB below the road. Call that point C.
Now, we want the height h = BT
Let
x = SC
y = CB
Now, we have
y/60 = sin 14°
x/60 = cos 14°
(h+y)/x = tan 55°
Combine all that to get
(h+60sin14°)cot55° = 60cos14°
I get h = 68.63 feet
To find the height of the telephone pole, we can make use of trigonometry and solve the problem step-by-step. Here's how you can go about it:
Step 1: Determine the opposite and adjacent sides of the right triangle formed:
Let's consider the right triangle in this scenario. The height of the telephone pole is the side opposite to the angle of elevation of the sun. The distance the shadow extends down the road is the side adjacent to the angle of elevation.
Step 2: Identify the appropriate trigonometric ratio:
Given that the angle of elevation is 55 degrees and the slope of the road is 14 degrees, we can see that the angle between the road and the height of the pole is 14 degrees (since the road slopes upward). Since we know the opposite and adjacent sides, we can utilize the tangent function, which relates the ratio of the opposite side to the adjacent side:
tan(angle) = opposite/adjacent
Step 3: Solve the equation:
Using the tangent function, we can now plug in the known values:
tan(14 degrees) = height of the pole / 60 feet
Solve for the height of the pole:
height of the pole = tan(14 degrees) * 60 feet
Step 4: Calculate the answer:
Using a scientific calculator or an online trigonometric calculator, evaluate the expression:
height of the pole ≈ 14.54 feet
Therefore, the height of the telephone pole is approximately 14.54 feet.