To solve this problem, we can use the principle of similar triangles.
Letβs denote:
- The height of the cell phone tower as \( h \).
- The height of the post as \( 4 \) feet \( 6 \) inches, which can be converted to feet. There are \( 12 \) inches in a foot, so \( 6 \) inches is \( \frac{6}{12} = 0.5 \) feet. Thus, the height of the post is \( 4 + 0.5 = 4.5 \) feet.
- The length of the shadow of the cell phone tower is \( 100 \) feet.
- The length of the shadow of the post is \( 3 \) feet \( 4 \) inches. Converting this to feet, \( 4 \) inches is \( \frac{4}{12} \approx 0.333 \) feet. So the shadow of the post is \( 3 + 0.333 = 3.333 \) feet.
By the property of similar triangles, we can set up the following proportion:
\[
\frac{h}{100} = \frac{4.5}{3.333}
\]
Now, we can solve for \( h \):
1. Cross-multiply:
\[
h \cdot 3.333 = 4.5 \cdot 100
\]
2. Calculate \( 4.5 \cdot 100 \):
\[
4.5 \cdot 100 = 450
\]
3. The equation now looks like this:
\[
h \cdot 3.333 = 450
\]
4. To find \( h \), divide both sides by \( 3.333 \):
\[
h = \frac{450}{3.333}
\]
5. Now, perform the division:
\[
h \approx 135
\]
Thus, the height of the cell phone tower is approximately **135 feet**.