A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake. She can bicycle at the rate of 6 mi/h and row a boat at 3 mi/h. What is the shortest amount of time she can travel to point C?
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math.stackexchange.com/questions/839014/optimization-problem-not-sure-how-to-proceed
To find the shortest amount of time for the woman to reach point C, we need to determine the most efficient way for her to travel around the lake.
Step 1: Calculate the circumference of the circular lake.
The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius.
In this case, the radius of the lake is 2 miles, so the circumference is C = 2π(2) = 4π miles.
Step 2: Calculate the time it takes to row the boat across the lake.
The woman's rowing speed is 3 mi/h, and she needs to travel the entire circumference of the lake (4π miles).
Therefore, the time it takes to row the boat is t1 = distance / speed = (4π miles) / (3 mi/h) = (4/3)π hours.
Step 3: Calculate the time it takes to bicycle around the lake.
The woman's biking speed is 6 mi/h, and she needs to travel the entire circumference of the lake (4π miles).
Therefore, the time it takes to bicycle is t2 = distance / speed = (4π miles) / (6 mi/h) = (2/3)π hours.
Step 4: Compare the times and determine the shortest amount of time.
The woman can either row the boat or bicycle around the lake. Comparing the times calculated in steps 2 and 3:
t1 = (4/3)π hours
t2 = (2/3)π hours
Since (2/3)π is smaller than (4/3)π, the shortest amount of time the woman can travel to point C is (2/3)π hours, which is approximately 2.1 hours.
To find the shortest amount of time the woman can travel to point C, we need to determine the most efficient route.
Here's how she can do it:
1. Assume that the woman will row her boat from point A to a point B somewhere along the edge of the lake.
2. From point B, she will bicycle straight across the diameter of the circle to reach point C.
3. She will then row her boat from point C back to the shore at point A.
Let's calculate the time for each leg of the journey:
1. Rowing from point A to point B:
- The distance from A to B is half the circumference of the lake, i.e., half the circumference of a circle with a radius of 2 mi.
- The distance is therefore π * 2 = 2π mi.
- The speed at which the woman rows is 3 mi/h.
- Thus, the time taken to row from A to B is (2π mi) / (3 mi/h) = (2/3)π h.
2. Bicycling from point B to point C:
- The distance from B to C is equal to the diameter of the circle, which is 2 * 2 = 4 mi.
- The speed at which the woman bicycles is 6 mi/h.
- Therefore, the time taken to bicycle from B to C is 4 mi / 6 mi/h = 2/3 h.
3. Rowing from point C back to point A:
- The distance from C to A is the same as the distance from A to B, which is 2π mi.
- The speed at which the woman rows is 3 mi/h.
- So, the time taken to row from C to A is (2π mi) / (3 mi/h) = (2/3)π h.
Now, let's add up the times for each leg of the journey:
(2/3)π h + (2/3) h + (2/3)π h
Since 2/3 and π are constants, we can combine them:
(2/3 + 2/3)π h = (4/3)π h
Therefore, the shortest amount of time the woman can travel to point C is (4/3)π hours, or approximately 4.189 hours, regardless of the specific location of point B along the shore of the lake.
Boat: V * t = 4 mi.
t = 4/V = 4/3 = 1.33 h.
Bicycle: The distance is 1/2 circle.
d = pi * 2r/2 = pi * r = 3.14 * 2 = 6.28 mi.
V * t = 6.28.
t = 6.28/V = 6.28/6 = 1.05 h. = Shortest time.