A 6 -ft vertical post casts a 16 -in shadow at the same time a nearby cell phone tower casts a 116 -ft shadow. How tall is the cell phone tower?

Question

UpStudy AI · Accepted Answer

Let the height of the cell phone tower be $$ h $$ feet.

1. **Convert the shadow length of the post from inches to feet:**

$$
   16 \text{ inches} = \frac{16}{12} = \frac{4}{3} \text{ feet}
   $$

2. **Set up the proportion using similar triangles:**

For the post, the ratio of height to shadow length is:

$$
   \frac{6}{\frac{4}{3}}
   $$

For the tower, the ratio is:

$$
   \frac{h}{116}
   $$

Because the triangles are similar, we have:

$$
   \frac{6}{\frac{4}{3}} = \frac{h}{116}
   $$

3. **Solve the proportion for $$ h $$:**

Simplify the left-hand side:

$$
   \frac{6}{\frac{4}{3}} = 6 \times \frac{3}{4} = \frac{18}{4} = \frac{9}{2}
   $$

Now, equate and solve:

$$
   \frac{9}{2} = \frac{h}{116} \quad \Longrightarrow \quad h = 116 \times \frac{9}{2}
   $$

Multiply:

$$
   h = 116 \times 4.5 = 522
   $$

4. **Final Answer:**

The cell phone tower is $$ 522 $$ feet tall.
The cell phone tower is 522 feet tall.

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