676 A flagpole casts a shadow 16.60 meters long. Tim stands at a distance of 12.45 meters from the base of the flagpole, such that the end of Tim's shadow meets the end of the flagpole's shadow. If Tim is 1.65 meters tall, determine and state the height of the flagpole to the nearest tenth of a meter.

To determine the height of the flagpole, we can use the concept of similar triangles. In this scenario, we have two triangles that are similar: one formed by the flagpole and its shadow, and the other formed by Tim and his shadow. The height of the flagpole can be calculated using the formula: \[ \frac{\text{Height of the flagpole}}{\text{Length of the flagpole's shadow}} = \frac{\text{Height of Tim}}{\text{Length of Tim's shadow}} \] Here, the length of the flagpole's shadow is 16.60 meters, and the length of Tim's shadow can be found by adding the distance Tim stands from the flagpole (12.45 meters) to the length of his own shadow. The key is to realize Tim's shadow ends at the tip of the flagpole's shadow, which implies: \[ \text{Length of Tim's shadow} = 16.60 - 12.45 = 4.15 \text{ meters} \] Now we can set up the proportion: \[ \frac{h}{16.60} = \frac{1.65}{4.15} \] Solving for \(h\) (height of the flagpole): \[ h = \frac{1.65 \times 16.60}{4.15} \] Calculating this gives: \[ h = \frac{27.39}{4.15} \approx 6.60 \text{ meters} \] Thus, the height of the flagpole is approximately **6.6 meters**.