Mathematics

Mathematics

Problem

A telescopic light pole is being extended so that its height increases at 0.2 ft/s0.2\text{ ft/s}. Simultaneously, a 66-ft tall skateboarder moves directly away from the base of the pole at 5 ft/s5\text{ ft/s}. When the pole is 18 ft18\text{ ft} tall and the skateboarder is 12 ft12\text{ ft} from the base, what is the speed of the tip of the skateboarder’s shadow? a) 5.0 ft/s5.0\text{ ft/s} b) 8.9 ft/s8.9\text{ ft/s} c) 7.4 ft/s7.4\text{ ft/s} d) 6.6 ft/s6.6\text{ ft/s}
Intuitive explanation

This is a related rates problem where two moving objects create a shadow whose tip moves along the ground—you need to track how fast that shadow tip travels. The key is to set up a similar triangles relationship: the tall pole and the skateboarder (with their respective shadows) form similar triangles, so their height-to-shadow ratios are equal. Once you write that constraint, differentiate with respect to time to connect the pole's growth rate, the skateboarder's walking speed, and the shadow tip's speed.

Intuitive explanation

This is a related rates problem where two moving objects create a shadow whose tip moves along the ground—you need to track how fast that shadow tip travels. The key is to set up a similar triangles relationship: the tall pole and the skateboarder (with their respective shadows) form similar triangles, so their height-to-shadow ratios are equal. Once you write that constraint, differentiate with respect to time to connect the pole's growth rate, the skateboarder's walking speed, and the shadow tip's speed.

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