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.

Need a picture:

Q

1 mile

2 miles

3 miles

x

3 - x

Find x to minimize time

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

To row: 42 x

To walk: 1)3( 2 x

The time to row is: milesxmiles

hour4

22

Notice I used unit analysis to find I had to divide by 2 and not multiply by 2

The time to walk is:

milesxmiles

hour1)3(

42

The primary equation is the total time:

1)3(4

14

2

1 22 xxT

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.

1)3(4

14

2

1 22 xxT

To minimize, find the derivative = 0

This will be good chain rule practice

))1)(3(2(1)3(

2

14124

2

121' 2

122

12 xxxxT

Outside Inside Outside Inside

Do you see another chain rule here?

))1)(3(2(1)3(

2

14124

2

121' 2

122

12 xxxxT

We need to solve this mess = 0

First, let’s clean it up a bit by multiplying constants

)3(1)3(4

1421' 2

122

12 xxxxT

Wow, let’s cheat and use MathPert to graph this and find the solution

Click on this

Now this

Select “Type it in”

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

click OK

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

Here is our graph, we want where it equals 0, ie, the x-intercept

click this about 4 times to get a better view

This looks like the point

Use the Point and Slope Tool

click it and center the crosshairs on the intercept

y = 0 and x = 1

this is where the derivative is equal to 0!

We just use a ‘numerical” technique to solve the equation

Just to make sure, evaluate at x = 1:

)3(1)3(4

1421' 2

122

12 xxxxT

You should get:

054

2

52

1

The man should aim for a point 1 mile down the coast

)3(1)3(4

1421' 2

122

12 xxxxT

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)