A woman wants to measure the height of a nearby tower. She places a 7 ft pole in the shadow of the tower so that the shadow of the pole is exactly covered by the shadow of the tower. The total length of the tower's shadow is 187 ft, and the pole casts a shadow that is 6.75 ft long. How tall is the tower? Round your answer to the nearest foot. (The figure is not drawn to scale.)

1 answer

To solve for the height of the tower, we can use similar triangles. The ratio of the height of an object to the length of its shadow is the same for both the pole and the tower.

Let \( h \) be the height of the tower. According to the problem:

  • The height of the pole is 7 ft.
  • The length of the pole's shadow is 6.75 ft.
  • The length of the tower's shadow is 187 ft.

Using the proportion, we have:

\[ \frac{\text{Height of Pole}}{\text{Shadow of Pole}} = \frac{\text{Height of Tower}}{\text{Shadow of Tower}} \]

Substituting the known values:

\[ \frac{7 \text{ ft}}{6.75 \text{ ft}} = \frac{h}{187 \text{ ft}} \]

Now, we can cross-multiply to solve for \( h \):

\[ 7 \text{ ft} \times 187 \text{ ft} = 6.75 \text{ ft} \times h \]

Calculating \( 7 \times 187 \):

\[ 7 \times 187 = 1309 \]

Thus, we have:

\[ 1309 = 6.75h \]

Now, divide both sides by 6.75 to find \( h \):

\[ h = \frac{1309}{6.75} \]

Calculating \( \frac{1309}{6.75} \):

\[ h \approx 193.15 \]

Rounding to the nearest foot, we find:

\[ h \approx 193 \]

Thus, the height of the tower is \( \boxed{193} \) ft.