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Q. 50

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Calculus
Found in: Page 860
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Illustration

Short Answer

Annie is conscious of tidal currents when she is sea kayaking. This activity can be tricky in an area south-southwest of Cattle Point on San Juan Island in Washington State. Annie is planning a trip through that area and finds that the velocity of the current changes with time and can be expressed by the vector function

0.4cosπ(t8)6,1.1cosπ(t11)6,

where t is measured in hours after midnight, speeds are given in knots and point due north.

(a) What is the velocity of the current at 8:00 a.m.?

(b) What is the velocity of the current at 11:00 a.m.?

(c) Annie needs to paddle through here heading southeast, 135 degrees from north. She wants the current to push her. What is the best time for her to pass this point? (Hint: Find the dot product of the given vector function with a vector in the direction of Annie’s travel, and determine when the result is maximized.)

Ans:

(a) The velocity of the current at 8:00 a.m is v(t)=0.4,0 .

(b) The velocity of the current at 11:00am is v(t)=0, -1.1

(c) The best time to pass her at the point is 11:00 AM

See the step by step solution

Step by Step Solution

Step 1. Given information.

given,

0.4cosπ(t8)6,1.1cosπ(t11)6,

Step 2. The objective of this problem is to find the velocity of tidal currents at a different time of the day at <0,1>due south.

The velocity of current changes with time and can be expressed as

v(t)=0.4cosπ(t8)6,1.1cosπ(t11)6

Step 3. (a)  Time is measured in hours after midnight. For the velocity of the current at 8:00 A.M 

total time t in 8 hours

Substitute t=8 in the expression of velocity.

localid="1649607446490" v(t)=0.4cosπ(88)6,1.1cosπ(811)6v(t)=0.4cos(0),1.1cosπ2 v(t)=0.4,0

Step 4. (b)   For the velocity of current at 11:00 AM

Total time t is 11 hours.

Substitute t=11 in the expression of velocity.

v(t)=0.4cosπ(118)6,1.1cosπ(1111)6v(t)=0.4cosπ2,1.1cos(0)v(t)=0,1.1

Step 5. (c)  Velocity in the direction of Annie's travel

r(t)=cos(45),sin(45)

For role="math" localid="1649607185923" v(t)=0.4,0

v(t)r(t)=0.4,0cos(45),sin(45)v(t)r(t)=0.4cos(45)=0.2828

For v(t)=0,-1.1

role="math" localid="1649607287767" v(t)r(t)=0,1.1cos(45),sin(45)v(t)r(t)=1.1sin(45)=0.7778 v(t)r(t) is maximum at 11:00 A.M.

Therefore, the best time to pass her at this point is 11:00 AM.

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