A radio tower stands on the top of a hill that is 85 feet high. From the bottom of the hill, the elevation angle to the base of the tower is 42 degrees and the elevation angle to the top of the tower is 60 degrees. How tall is the tower?
I would like to know what steps to take to arrive at this answer.
Step 1:
Draw a diagram, with a tower of height h on top of a hill 85' hight.
The 42° angle of elevation is measured at a horizontal distance x from the base of the tower.
Step 2:
Calculate x knowing that tan(42)=85/x
Step 3:
Calculate h knowing that
tan(60)=(85+h)/(x)
Step 4:
Solve for h.
To determine the height of the tower, we can use trigonometry, specifically the tangent function. Here are the steps:
1. Draw a diagram: Visualize the scenario to better understand the given information. Draw a right triangle with the hill as the base, the tower as its vertical side, and the line of sight from the bottom of the hill to the top of the tower as the hypotenuse.
2. Label the diagram: Mark the height of the hill (85 ft), the elevation angle to the base of the tower (42 degrees), and the elevation angle to the top of the tower (60 degrees). Label the unknown height of the tower as 'x.'
3. Identify the trigonometric ratios: Since we are dealing with angles and sides, we'll be using the tangent function since it relates the opposite side to the adjacent side in a right triangle. In this case, we want to find the height of the tower, which is the opposite side, and we know the angle and the adjacent side (the height of the hill).
4. Write down the tangent equation: The equation for tangent is tangent(angle) = opposite/adjacent. Plugging in the values for the angle to the base of the tower (42 degrees) and the adjacent side (85 ft), the equation becomes: tangent(42) = x/85.
5. Solve the equation: Rearrange the equation to isolate 'x.' Multiply both sides by 85 to get rid of the denominator: x = 85 * tangent(42).
6. Use a calculator to evaluate tangent(42): The tangent of an angle can be calculated using a scientific calculator or an online calculator. Plug in '42' and calculate the tangent, which is approximately 0.9004.
7. Calculate x, the height of the tower: Multiply 85 by the calculated tangent value: x = 85 * 0.9004.
8. Simplify x to find the height of the tower: Use a calculator to calculate 85 multiplied by 0.9004. The answer is approximately 76.238 feet.
Therefore, the height of the tower is approximately 76.238 feet.