Answer: To minimize the time taken by the woman to reach point C, she should minimize the total distance traveled, which is the sum of the distance she walks and the distance she rows.
Let's call point B the point where the woman switches from walking to rowing. We can find the location of point B by drawing a straight line from A to the center of the lake, and then continuing that line on the other side of the lake to point C. Point B is the point where this line intersects the circle of the lake.
Since the radius of the lake is 4, the distance from A to the center of the lake is also 4. Therefore, the distance from A to B is also 4. The distance from B to C is also 4, since C is diametrically opposite to A.
Let's call the distance that the woman rows from B to C d. Then the distance that she walks from A to B is 4 - d.
The time taken to walk a distance of (4 - d) miles is:
t1 = (4 - d) / 10
The time taken to row a distance of d miles is:
t2 = d / 5
The total time taken is:
T = t1 + t2 = (4 - d) / 10 + d / 5
Simplifying, we get:
T = (8 + d) / 20
To minimize T, we need to find the value of d that minimizes (8 + d) / 20. We can do this by taking the derivative of (8 + d) / 20 with respect to d and setting it to 0:
d(T) / d(d) = 1/20
Setting this to 0, we get:
1/20 = 0
This is obviously not true, so there is no minimum value of T. However, we can see that as d gets larger, T gets larger, and as d gets smaller, T gets smaller. Therefore, the minimum value of T occurs at one of the endpoints of the interval [0, 4]. Since d cannot be negative, the only endpoint we need to consider is d = 4.
When d = 4, the woman rows the entire distance from B to C, and does not need to walk at all. Therefore, the total time taken is:
T = (8 + 4) / 20 = 0.6 hours
Therefore, the woman should walk to point B, and then row the rest of the way to point C, to arrive in the shortest possible time.
Explanation: