By holding a ruler 3 feet in front of her eyes as illustrated in the diagram, Carla sees that the top and bottom points of a vertical cliff face line up with marks separated by 3.5" on the ruler. According to the map, Carla is about a half mile from the cliff. What is the approximate height of the cliff rounded to the nearest foot? Recall that a mile is 5280 feet.
To find the approximate height of the cliff, we need to use similar triangles.
First, let's establish the relationship between the ruler and the distance. We are given that the marks on the ruler are separated by 3.5 inches and Carla is holding the ruler 3 feet in front of her eyes. This forms a right triangle with the ruler, her eyes, and the cliff.
Since we are dealing with similar triangles, we can set up a proportion. The small triangle formed by the ruler and her eyes has a height of 3 feet and a base of 3.5 inches. The larger triangle formed by the ruler, her eyes, and the cliff has an unknown height h and a base of 0.5 miles (which is equal to 2640 feet).
Using the proportion:
(height of small triangle) / (base of small triangle) = (height of large triangle) / (base of large triangle)
Plugging in the given values:
3 ft / 3.5 in = h / 2640 ft
To get rid of the inches, we need to convert the measurements to the same unit. There are 12 inches in 1 foot, so:
3 ft / (3.5 in * 12 in/ft) = h / 2640 ft
Now, let's solve for h:
3 ft / (3.5 * 12) = h / 2640 ft
(3 ft * 2640 ft) / (3.5 * 12) = h
(7920 ft^2) / (42 in) = h
7920 ft^2 / 42 = h
188.57 ft = h
Rounding to the nearest foot, the approximate height of the cliff is 189 feet.