To solve the problem of finding the height of the telephone pole using the given dimensions, we can use the principle of similar triangles. Here's a step-by-step solution:
1. Understand the Given Information:
- The yardstick casts a shadow of 24 inches.
- The height of the yardstick is 36 inches.
- The telephone pole casts a shadow of 20 feet and 8 inches.
2. Convert the Shadow of the Telephone Pole to Inches:
- First, convert the feet to inches: [tex]\( 20 \, \text{ft} = 20 \times 12 \, \text{in} = 240 \, \text{in} \)[/tex].
- Add the remaining inches: [tex]\( 8 \, \text{in} \)[/tex].
- Total shadow length of the telephone pole: [tex]\( 240 \, \text{in} + 8 \, \text{in} = 248 \, \text{in} \)[/tex].
3. Set Up the Proportion Using Similar Triangles:
- Since the yardstick and the telephone pole form similar triangles with their shadows, we can write the proportion as:
[tex]\[
\frac{\text{Height of Yardstick}}{\text{Shadow of Yardstick}} = \frac{\text{Height of Telephone Pole}}{\text{Shadow of Telephone Pole}}
\][/tex]
- Substitute the known values:
[tex]\[
\frac{36 \, \text{in}}{24 \, \text{in}} = \frac{\text{Height of Telephone Pole}}{248 \, \text{in}}
\][/tex]
4. Solve for the Height of the Telephone Pole:
- Cross-multiply to find the unknown height:
[tex]\[
36 \times 248 = 24 \times \text{Height of Telephone Pole}
\][/tex]
[tex]\[
36 \times 248 = 8928 \, \text{in}^2
\][/tex]
- Divide both sides by 24:
[tex]\[
\text{Height of Telephone Pole} = \frac{8928 \, \text{in}^2}{24 \, \text{in}} = 372 \, \text{in}
\][/tex]
5. Round to the Nearest Inch:
- The calculation gives us 372 inches, which is already an integer, so there is no need to round.
Thus, the height of the telephone pole, to the nearest inch, is [tex]\( \boxed{372} \, \text{in} \)[/tex]. Therefore, the correct answer is:
C. 372 in.
