A woman at a point A on the shore of a circular lake with radius r=4 wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 10mph and row a boat at 5mph. What is the shortest amount of time it would take her to reach point C?

I'm not sure where to start... :(

Attached is a drawing to set up the problem, hopefully that helps.
The red lines are tracing the rowing and walking to get to other side.
dR = distance rowed
dw = distance walked
x = the angle in which to start rowing initially

dR can be found by using Law of cosines
\(dR = \sqrt{4^2 +4^2 - 2(4)(4) cos(180-2x)} = 4\sqrt{2}\sqrt{1-cos(180-2x)}
\)
dw is arc length, which is just radius*arc
\(dW = 4*2x = 8x\)

next we need t set up a function for Time in terms of x.
This is so we can minimize the time it takes to get to other side.
Time = distance/rate

\(T = \frac{dR}{5} + \frac{dW}{10} = \frac{4 \sqrt{2}}{5} \sqrt{1-cos(180-2x)}+\frac{4}{5} x\)
Finally, take derivative and set equal to 0
Once you solve for x, plug it back into Time function to obtain final answer.

olayemiolakunle65 olayemiolakunle65 · Answer 2 · 2021-11-23T11:44:38-05:00

The diameter of a circle is a line passing through its center which divides the circle into two equal parts. The radius is half of a diameter. Thus, the shortest amount of time taken is 0.8 hours.

A lake is a body of flowing water which is not a wide/ large as a sea or an ocean. Given that the lake has a circular shape, then its radius is half of its diameter.

So that the diameter of the lake is twice its radius. This can be determined as:

D = 2r

= 2 x 4

D = 8 m

The diameter of the lake is 8 meters.

Given that:

i. She can walk at the rate of 10 mph.

ii. She can row a boat at 5 mph.

The speed of an object is the rate of distance moved to the time taken.

i.e speed = \(\frac{distance}{time}\)

Thus, to determine the shortest time taken, then:

time = \(\frac{distance}{speed}\)

a. If she walks diametrically to point C, the time taken would be;

time = \(\frac{8m}{10mph}\)

= 0.8 h

The time taken in this case is 0.8 hours.

b. If she row a boat diametrically to point C, the time taken would be;

time = \(\frac{8m}{5mph}\)

= 1.6 h

The time taken in this case is 1.6 hours.

Therefore, the shortest time it would take her to reach point C is 0.8 hours.

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