Explanation Given: Let ‘ r’ denote the speed along the road, and ‘h’ denote speed along highway. Concept Used: For finding critical point of a given function, we’ll find the derivative with respect to that given variable of that function. Then make this derivative equal to zero. Calculation: Formula to calculate speed. Speed= distance time Perpendicular distance from Ranch to point P is 30km. Distance from point P to city is 50km. The distance from Ranch to the point Q = 30 2 + x 2 km and the distance from point Q to the city = 50 − x km. The speed along road while traveling distance Q = 30 2 + x 2 is r km/h and the speed along the city while traveling 50 - x km from point Q is h km/h. Formula to calculate time. time= distance Speed T ( x ) = 30 2 + x 2 r + 50 − x h Now, solve for critical point x. Differentiate above function T( x ) with respect to x. T ' ( x ) = 1 r ( 1 2 ( 30 2 + x 2 ) − 1 2 2 x ) − 1 h T ' ( x ) = x r ( 30 2 + x 2 ) 1 2 − 1 h T ' ( x ) = x r ( 30 2 + x 2 ) − 1 h Put T ' ( x ) = 0 , x r ( 30 2 + x 2 ) − 1 h = 0 x r ( 30 2 + x 2 ) = 1 h x h = r ( 30 2 + x 2 ) Take square on both sides To determine (b) To explain that there cannot be a critical point at x = 0, but depending on the speeds, critical point can be arbitrary close to 0.

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Question

Chapter 4.7, Problem 19E

To determine

(a)

To show that the travel-time function T ( x ) has critical point at x=30(h/r)2−1.

To determine

(b)

To explain that there cannot be a critical point at x = 0, but depending on the speeds, critical point can be arbitrary close to 0.

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Chapter 4.7, Problem 18E

Chapter 4.7, Problem 20E

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Chapter 4 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

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(a) To show that the travel-time function T ( x ) has critical point at x = 30 ( h / r ) 2 − 1 .

Solution Summary: The author illustrates how the travel-time function T ( x ) has a critical point.

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(a)

To show that the travel-time function T ( x ) has critical point at x=30(h/r)2−1.

(b)

To explain that there cannot be a critical point at x = 0, but depending on the speeds, critical point can be arbitrary close to 0.

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Chapter 4 Solutions

(a) To show that the travel-time function T ( x ) has critical point at x = 30 ( h / r ) 2 − 1 .

Solution Summary: The author illustrates how the travel-time function T ( x ) has a critical point.

Concept explainers

Videos

(a) To show that the travel-time function T ( x ) has critical point at x=30(h/r)2−1.

(b) To explain that there cannot be a critical point at x = 0, but depending on the speeds, critical point can be arbitrary close to 0.

Want to see the full answer?

Chapter 4 Solutions

(a)

To show that the travel-time function T ( x ) has critical point at x=30(h/r)2−1.

(b)

To explain that there cannot be a critical point at x = 0, but depending on the speeds, critical point can be arbitrary close to 0.