I) A Light is on Top of a 12 foot pole and an 5 fT 6 in tall person is walking away from the pole at a rate of \( 2 \mathrm{fT} / \mathrm{sec} \) a) What rate is the Tip of the shadow moving away from the pole when the person is 25 fT from the pole? b) AT what rate is the Tip of the shadow moving away from the person when the person is 25 fT from the pole?
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To tackle this problem, it's helpful to imagine the scenario visually. When the person stands next to the pole, the height of the pole and the person, along with their distance from each other, creates similar triangles. This gives us a relation to find the length of the shadow and how fast it's changing as the person walks away. At the moment the person is 25 ft away, we can use similar triangles to derive the rates. For part a, you’ll find that the tip of the shadow moves at a rate of about 2.88 ft/sec as the person moves away from the pole. For part b, you would need to calculate the difference between the rates of the shadow and the person's movement, which results in the tip of the shadow moving away from the person at a rate of approximately 0.88 ft/sec. In scenarios like this, using the derivatives of similar triangles can bring clarity! As you solve these kinds of problems, a common mistake is forgetting to account for the full lengths involved, ensuring you're measuring from both the pole and the person accurately. Keeping a diagram can help visualize the relationships and avoid such pitfalls!