A woman at point A on the shore of a circular lake with radius 2.5mi 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 a rate of 3.25 mi/hr and row at a rate of 2 mi/hr.
Find shortest time.
If she walks to subtend an angle of θ, she walks 2.5θ miles
The distance to row across is thus
d^2 = 2(2.5^2)(1+cosθ)
time elapsed is
t = 2.5θ/3.25 + 2.5√2(1+cosθ)/2
= .7692θ + 1.7678√(1+cosθ)
dt/dθ = .7629 - .8839sinθ/√(1+cosθ)
we want dt/dθ=0. That happens when θ=1.31
So, she walks 2.5*1.31 = 3.275 mi
and she rows 3.965 mi
time = 3.275/3.25 + 3.965/2 = 3 hours
For anyone looking at the response, I think it should be 1-cosθ, not 1+cosθ.
To find the shortest time for the woman to reach point C from point A, we need to find the quickest combination of walking and rowing.
Let's break down the problem into smaller steps:
Step 1: Find the distance between points A and C
The diameter of the circular lake is twice the radius, so the diameter is 2 * 2.5mi = 5mi. Therefore, the distance between points A and C is 5mi.
Step 2: Calculate the time it takes to row across the lake
Since the woman can row at a rate of 2 mi/hr, the time it takes to row across the lake is distance/row speed = 5mi/2 mi/hr = 2.5 hours.
Step 3: Calculate the time it takes to walk along the circumference of the lake
The circumference of a circle is given by 2 * π * radius. In this case, it is 2 * 3.14 * 2.5mi = 15.7mi.
Therefore, the time it takes to walk along the circumference is distance/walking speed = 15.7mi/3.25 mi/hr ≈ 4.83 hours.
Step 4: Compare the times from Step 2 and Step 3 and choose the shorter time
The time it takes to row across the lake (2.5 hours) is shorter than the time it takes to walk along the circumference (4.83 hours).
Thus, the shortest possible time for the woman to reach point C from point A is 2.5 hours by rowing across the lake.
To find the shortest time for the woman to travel from point A to point C, we need to calculate the time for both walking and rowing, and choose the option that takes less time.
Let's consider two scenarios:
1. Walking all the way around the lake.
2. Rowing across the lake directly.
Scenario 1: Walking all the way around the lake
The circumference of a circle is given by the formula: C = 2πr
In this case, the circumference of the lake is 2π * 2.5 miles.
To find the time to walk around the lake, we divide the distance by the walking rate:
Time to walk = (Circumference of the lake) / (Walking rate)
Scenario 2: Rowing across the lake directly
We need to find the distance from point A to point C, which is equal to the diameter of the lake. The diameter of a circle is twice the radius.
Distance from A to C = 2 * 2.5 miles
To find the time to row across the lake, we divide the distance by the rowing rate:
Time to row = (Distance from A to C) / (Rowing rate)
Now we can calculate the time for both scenarios and compare them to find the shortest time.
Time to walk around the lake = (2π * 2.5 miles) / (3.25 mi/hr)
Time to row across the lake = (2 * 2.5 miles) / (2 mi/hr)
Calculating these values will give us the shortest time.