If a 36 inch yardstick casts a 21 foot shadow, how tall is a building whose shadow is 168 feet? What is the scale factor
The Correct Answer and Explanation is:
We can use the concept of similar triangles to solve this problem. The yardstick and the building form two similar right triangles, where the height of the yardstick and the building’s height are the corresponding sides, and the length of the shadows are the other corresponding sides.
Step 1: Set up the proportion
Let’s denote the height of the building as hhh. The length of the yardstick’s shadow is given as 21 feet, and the shadow of the building is 168 feet. The height of the yardstick is 36 inches, which is equivalent to 3 feet (since 12 inches = 1 foot).
The proportion between the height of the yardstick and the building can be set up as follows:Height of YardstickLength of Yardstick Shadow=Height of BuildingLength of Building Shadow\frac{\text{Height of Yardstick}}{\text{Length of Yardstick Shadow}} = \frac{\text{Height of Building}}{\text{Length of Building Shadow}}Length of Yardstick ShadowHeight of Yardstick=Length of Building ShadowHeight of Building
Substituting the given values:321=h168\frac{3}{21} = \frac{h}{168}213=168h
Step 2: Solve for hhh
Now, cross-multiply to solve for the height of the building:3×168=21×h3 \times 168 = 21 \times h3×168=21×h504=21h504 = 21h504=21hh=50421=24 feeth = \frac{504}{21} = 24 \text{ feet}h=21504=24 feet
So, the height of the building is 24 feet.
Step 3: Find the scale factor
The scale factor is the ratio of the height of the building to the height of the yardstick. From the previous calculation, we know:Scale Factor=Height of BuildingHeight of Yardstick=243=8\text{Scale Factor} = \frac{\text{Height of Building}}{\text{Height of Yardstick}} = \frac{24}{3} = 8Scale Factor=Height of YardstickHeight of Building=324=8
Conclusion:
The height of the building is 24 feet, and the scale factor is 8. This means the building is 8 times taller than the yardstick. The shadows of the two objects are also proportional, with the building’s shadow being 8 times longer than the yardstick’s shadow.
