RC 7-4: Similar Triangles Shortcuts and Proof A 10-ft vertical post casts a 20 -in shadow at the same time a nearby cell phone tower casts a 120 -ft shadow. How tall is the cell phone tower? The cell phone tower's height is \( \square \mathrm{ft} \). (Simplify your answer.)

To find the height of the cell phone tower, we can use the property of similar triangles, which states that the ratios of corresponding sides are equal. The ratio of the height of the post to the length of its shadow will be the same as the ratio of the height of the cell phone tower to the length of its shadow. Let \( h \) be the height of the cell phone tower. We set up the proportion: \[ \frac{10 \text{ ft}}{20 \text{ in}} = \frac{h}{120 \text{ ft}} \] First, we need to convert 20 inches to feet because the other measurements are in feet. There are 12 inches in a foot, so: \[ 20 \text{ in} = \frac{20}{12} \text{ ft} = \frac{5}{3} \text{ ft} \] Now, substituting this back into our proportion gives: \[ \frac{10 \text{ ft}}{\frac{5}{3} \text{ ft}} = \frac{h}{120 \text{ ft}} \] Cross-multiplying to solve for \( h \): \[ 10 \cdot 120 = h \cdot \frac{5}{3} \] \[ 1200 = \frac{5h}{3} \] To isolate \( h \), multiply both sides by 3: \[ 3600 = 5h \] Now divide by 5: \[ h = \frac{3600}{5} = 720 \text{ ft} \] The height of the cell phone tower is \( 720 \text{ ft} \).