SOLUTION: Emma launches her boat from point A on a bank of a straight river, 3km wide, and wants to reach Point B (8km downstream on the opposite bank) as quickly as possible.
She could
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She could
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Question 1137355: Emma launches her boat from point A on a bank of a straight river, 3km wide, and wants to reach Point B (8km downstream on the opposite bank) as quickly as possible.
She could proceed in any of three ways:
- (a) Row her boat directly across the river to Point C and then run to B
- (b) Row directly to B
- (c) Row to some point D between C and B, and then run to B
If she can row 6km/h and run 8km/h, where should she land to reach B as soon as possible?
Note: We assume that the speed of the water is negligible compared with the speed at which she rows. Found 2 solutions by mananth, ikleyn:Answer by mananth(16946) (Show Source):
You can put this solution on YOUR website! Emma launches her boat from point A on a bank of a straight river, 3km wide, and wants to reach Point B (8km downstream on the opposite bank) as quickly as possible.
- Row her boat directly across the river to Point C and then run to B Time = distance / speed. Time AC + time CB = total time
Time taken will be 3/6 + 8/8 = 1.5 hours
- Row directly to B
The hypotenuse distance will be
time taken for this distance will be = 1.42 hours
- Row to some point D between C and B, and then run to B
Obviously this will take more time since rowing distance is greater than running distance.
While I agree with the @mananth's answers to (a) and (b), I totally disagree with his answer to (c), which is incorrect.
Let x be the "x-coordinate" of the intermediate "target" point D between points C and B on the opposite bank of the river.
Then the total time t(x) to get from A to B is
t(x) = + hours. (1)
The plot of this function is shown in the figure below, and it clearly shows that the function t(x) has a minimum between 0 and 8.
Plot y = +
To find the value of x which provides the minimum to t(x), take the derivative of t(x)
t'(x) = - ,
equate it to zero and solve the obtained equation for x
- = 0,
8x =
64x^2 = 36(9+x^2)
64x^2 = 36*9 + 36x^2
64x^2 - 36x^2 = 324
x^2 = = 11.571
x = = 3.4 (approximately).
Answer. The target point to minimize time is 3.4 kilometers from C to B,
giving time t(3.4) = + = 1.331 hours.
