A woman wants to measure the height of a nearby tower. She places a pole in the shadow of the tower so that the shadow of the pole is exactly covered by the shadow of the tower. The distance between the pole and the tower is , and the pole casts a shadow that is long. How tall is the tower? Round your answer to the nearest foot.
To find the height of the tower, we can set up a proportion using similar triangles. Here's how you can do it:
1. Assign variables: Let's call the height of the tower "h" and the length of the shadow of the tower "s".
2. Note that the pole, the shadow it casts, and the tower create two similar right triangles.
3. Set up the proportion: The ratio of the height of the tower to its shadow should be the same as the ratio of the height of the pole to its shadow:
h / s = x / d
Since we are looking for "h", leave it as a variable. The height of the pole is "x".
4. Plug in the known values: You are given the length of the pole's shadow (d) and the height of the pole (x), so you can substitute these values into the proportion:
h / s = x / d
h / s = x / 18
5. Solve for "h": Cross-multiply and solve for "h":
h * d = s * x
h = (s * x) / d
Substituting the given values, you get:
h = (42 * s) / 18
6. Calculate the height of the tower: Now, substitute the given values of the pole's shadow length (s) and the distance between the pole and the tower (d):
h = (42 * 18) / 18
h = 42
The height of the tower is 42 feet.