A person stands at a distance from the tower and eyes the top of the tower at 30 degree angle of elevation. He then walks 425 feet towards the tower and eyes the top of the tower at a 58 degree angle of elevation. The person is 5 feet tall. How tall is the tower?
solve the problem for a person of height zero, then add 5 feet to the answer. Also assume his eyes are on the top of his head.
If he starts at distance x, and the tower has height h, then
h/x = tan 30
h/(x-425) = tan 58
eliminate x to get
h/tan30 = h/tan58 + 425
now just solve for h.
This is easy problem if you draw the triangles properly.
If we consider the height of the tower above the person to be x and the distance remaining to reach the tower from where he made the last measurement to be y then
tan 32 = y/x
0.625 = y/x
x = y/0.625 = 1.6 y
but we also know from the larger triangle (the starting point) that
tan 30 = x/(425+y)
0.577 = x/(425+y)
x = 245.37 + 0.577 y
substitute from first equation into second one
1.6 y = 245.37 + 0.577 y
1.023 y = 245.37
y = 245.37/1.023= 239.85 feet
x = 1.6 * 239.85 = 383.76 feet
the height of the tower = 5 + 383.76= 388.76 feet
Thank you so much for the help.
To find the height of the tower, we can use trigonometry. Let's break down the problem step by step.
Step 1: Find the distance between the initial position and the tower.
- The person first stands at a distance from the tower and eyes the top of the tower at a 30-degree angle of elevation.
- We can use basic trigonometry to find this distance. The opposite side of the angle represents the height, and the adjacent side represents the distance.
- Since the angle is 30 degrees and we know the opposite side is the person's height of 5 feet, we can use the tangent function to find the distance.
- The formula for tangent is tan(angle) = opposite/adjacent. Plugging in the values, we have tan(30) = 5/adjacent.
- Rearranging the equation to solve for the adjacent side (distance), we have adjacent = 5/tan(30).
Step 2: Calculate the height of the tower using the distance found in Step 1.
- The person then moves 425 feet towards the tower and sights the top of the tower at a 58-degree angle of elevation.
- Now, we can similarly use trigonometry to find the height.
- The adjacent side represents the new distance between the person and the tower, and the opposite side represents the additional height.
- Since we moved 425 feet, we add this distance to the previous distance found in Step 1.
- Using the formula we used in Step 1, the new distance is 425 + 5/tan(30).
- Using the tangent function again, we can set up the equation as tan(58) = height/(425 + 5/tan(30)).
- Rearranging the equation to solve for the height of the tower, we have height = tan(58) * (425 + 5/tan(30)).
Step 3: Evaluate the height of the tower.
- Finally, substitute the values into the equation to find the height of the tower.
- Use a calculator to calculate tan(58), tan(30), and evaluate the expression for height.
So, the height of the tower can be calculated as height = tan(58) * (425 + 5/tan(30)), where tan(58), tan(30), and the result of the expression should be evaluated using a calculator.