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Published byLina Rackliffe Modified about 1 year ago

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This is problem 49 page 219: A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located 3 miles down the coast and 1 mile inland. If he can row at 2 miles per hour and walk at 4 miles per hour, toward what point on the coast should he row in order to reach point Q in the least amount of time.

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Need a picture: Q 1 mile 2 miles 3 miles x 3 - x Find x to minimize time

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Use the Pythagorean theorem twice to find the distance to row and walk To row: To walk:

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The time to row is: Notice I used unit analysis to find I had to divide by 2 and not multiply by 2 The time to walk is:

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The primary equation is the total time: We don’t need a “secondary equation”. Why? The primary equation only has one variable, there is no need to substitute for a second variable.

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To minimize, find the derivative = 0 This will be good chain rule practice OutsideInside OutsideInside Do you see another chain rule here?

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We need to solve this mess = 0 First, let’s clean it up a bit by multiplying constants Wow, let’s cheat and use MathPert to graph this and find the solution

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Click on this

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Now this

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Select “Type it in”

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note the use of ( ) everywhere and the use of negative exponents now click OK

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Make sure this is the equation we want and it is. Click OK again

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Here is our graph, we want where it equals 0, ie, the x-intercept click this about 4 times to get a better view

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This looks like the point

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Use the Point and Slope Tool click it and center the crosshairs on the intercept

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y = 0 and x = 1 this is where the derivative is equal to 0! We just use a ‘numerical” technique to solve the equation

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Just to make sure, evaluate at x = 1: You should get: The man should aim for a point 1 mile down the coast

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To review what I typed into MathPert: 1/2((x^2+4)^(-1/2)(x))-1/4((3-x)^2+1)(-1/2)(3-x)) “Count” the number of opening (( versus the number of closing )), they must match in any expression. I could also type, which is easier: x/2sqrt(x^2+4)-(3-x)/4sqrt((3-x)^2+1)

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