Explanation Given: The cabin of the hunter is located 19 miles north and 8 miles west from the point along the river bank. The speed of the hunter along the river bank is 5 mph. The speed of the hunter on the rocky land is 2 mph. Calculation: Let 8 − x be the distance that the hunter travel along the river bank. Then x is the distance that he does not travel along river bank. The cabin is at 19 miles north. From Figure 1, it is observed that triangle APB is a right angled triangle. Applying Pythagorean Theorem for the given figure, The distance that he travel on land is 19 2 + x 2 = 361 + x 2 The formula for time taken is t = d r where d is distance and r is rate of speed. The time taken by him to travel along the river is 8 − x 5 The time taken by him to travel on the land is 361 + x 2 2 The total time is, T ( x ) = 8 − x 5 + 361 + x 2 2 = 8 5 − 1 5 x + 1 2 ( 361 + x 2 ) 1 2 Find the critical points by evaluating T ′ ( x ) and equating to 0. T ′ ( x ) = − 1 5 + 1 2 ( 1 2 ) ( 2 x ) ( 361 + x 2 ) − 1 2 = − 1 5 + x 2 ( 361 + x 2 ) − 1 2 Equate T ′ ( x ) to 0. − 1 5 + x 2 ( 361 + x 2 ) − 1 2 = 0 x 2 ( 361 + x 2 ) − 1 2 = 1 5 x 2 = ( 361 + x 2 ) 1 2 5 x = 2 (

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 6.2, Problem 50E

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To find: The distance that the hunter should travel along the river in order to reach his cabin in minimum time.

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Chapter 6.2, Problem 49E

Chapter 6.2, Problem 51E

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Calculus with Applications (11th Edition)

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