Height of a Lighthouse. The Cape Hatteras Lighthouse at the Outer Banks of North Carolina is the tallest lighthouse in North America. If a 5-foot woman casts a -foot shadow and the lighthouse casts a 48-foot shadow, approximately how tall is the Cape Hatteras Lighthouse?
The Correct Answer and Explanation is :
To solve this problem, we will use the concept of similar triangles. Since both the woman and the lighthouse cast shadows of different lengths, but the angle of elevation from the tip of the shadow to the top of each object is the same, their heights and shadow lengths are proportional. We can set up a proportion to solve for the height of the lighthouse.
Step-by-step solution:
- Understand the setup:
- The woman’s height is 5 feet.
- The length of the woman’s shadow is not provided, but it can be deduced as proportional to the height.
- The lighthouse’s shadow length is 48 feet.
- We need to determine the height of the lighthouse, denoted as ( h ).
- Set up the proportion:
We can use the property that the height of an object is proportional to the length of its shadow. Therefore, the proportion between the woman’s height and her shadow is the same as the proportion between the lighthouse’s height and its shadow. The formula for this is: [
\frac{\text{Height of woman}}{\text{Shadow of woman}} = \frac{\text{Height of lighthouse}}{\text{Shadow of lighthouse}}
] Substituting the known values: [
\frac{5}{\text{Shadow of woman}} = \frac{h}{48}
] The shadow of the woman is not provided explicitly, but we can assume that the shadow length of the woman and the lighthouse should be proportional. Therefore, solving for the height ( h ) will involve calculating the scale factor between the shadows. - Calculate the lighthouse height:
Now we can calculate the lighthouse height based on the proportions. ( h = 5 * ( \text{scale factor})