To determine To find: The path that takes the minimum time to reach the tree. Explanation Given: The total trail distance is 300m east along the trail and 800m north through the wood. The running speed of the man is 160m per minute along the trail and 40m per minute through the wood. Calculation: From the figure, x is the distance that the professor runs through the wood and 300 − x is the distance he runs along the trail. Applying Pythagorean theorem to the figure, The distance he runs through the wood is 800 2 + x 2 Here there are two separate times for two different paths and speeds. Time is the ratio of distance and speed. Time taken by him to run along the trail is 300 − x 160 Time taken by him to run through the wood is 800 2 + x 2 40 So, the total time is T ( x ) = 300 − x 160 + 800 2 + x 2 40 Now, we find the domain of the function T. The variable x should be non-negative. So, x ≥ 0 The function T is also non-negative since time cannot be negative. T ( x ) ≥ 0 300 − x 160 + 800 2 + x 2 40 ≥ 0 since x ≥ 0 , the second part of the above equation is always greater than zero. The first part must also be greater than zero. 300 − x 160 ≥ 0 300 − x ≥ 0 300 ≥ x So, the domain of T is [ 0 , 300 ] For finding the critical points of T, first we find d T d x and solve the equation d T d x = 0 for x . d T d x = d d x ( T ) = d d x ( 300 − x 160 + 800 2 + x 2 40 ) = d d x ( 300 − x 160 ) + d d x ( 800 2 + x 2 40 ) = − 1 160 + 1 40 ( 1 2 800 2 + x 2 ) ( 2 x ) On solving the above equation further, d T d x = − 1 160 + 1 40 ( x 800 2 + x 2 ) On solving the equation d T d x = 0 − 1 160 + 1 40 ( x 800 2 + x 2 ) = 0

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

11th Edition

ISBN: 9780133886832

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

Publisher: PEARSON

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Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

Ch. 6.1 - YOUR TURN 1 Find the absolute extrema of the...Ch. 6.1 - Prob. 2YT Ch. 6.1 - Prob. 1WE Ch. 6.1 - Prob. 2WE Ch. 6.1 - Prob. 1E Ch. 6.1 - Find the locations of any absolute extrema for the...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

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