Question

{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1

Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>

Accepted Answer

Understand the problem: We have a pursuit curve problem where a man is walking towards a dog. The dog moves north at 1 mi/hr from a crossroads, and the man starts 1 mile east of the crossroads, walking at a speed greater than 1 mi/hr directly towards the dog. Identify the function: The path followed by the man is given by the function y = f(x) = (s/2) * ((x^(s+1)/s) / (s+1) - (x^(s+1)/s) / (s-1)) + s / $$(s^2 - 1). $$This function describes the man's path in the xy-plane as he pursues the dog. Choose values for s: To analyze the pursuit curve, select various values of s greater than 1. For example, you might choose s = 1.5, s = 2, and s = 3. These values will help you observe how the curve changes as the man's speed increases. Graph the pursuit curve: For each chosen value of s, plot the function y = f(x) on a graph. This will visually represent the path the man takes as he pursues the dog. Use graphing software or a graphing calculator to assist with this step. Analyze the changes: As you increase the value of s, observe how the pursuit curve changes. Comment on the differences in the path's shape and direction. Typically, as s increases, the man's path becomes more direct and less curved, indicating a faster pursuit towards the dog.

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