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... :(

To determine the shortest amount of time it would take for the woman to reach point C, we need to compare the time taken if she walks along the shore with the time taken if she rows across the lake.

Let's start by calculating the time taken if she walks along the shore.

1. The perimeter of a circle is given by the formula: P = 2πr.
In this case, the perimeter of the lake is P = 2π(4).
Therefore, the distance the woman needs to walk along the shore to reach point C is equal to the circumference of the lake, which is 2π(4) or 8π miles.

2. The woman can walk at a rate of 10mph.
To calculate the time taken to walk along the shore, we divide the distance by the rate: Time = Distance / Rate.
Therefore, the time taken to walk along the shore is 8π / 10 or (4π / 5) hours.

Next, let's calculate the time taken if she rows across the lake.

1. The diameter of the lake is equal to twice the radius, which is 2r = 2(4) or 8 miles.
Since point C is diametrically opposite A, the woman needs to row across the lake along its diameter, which is 8 miles.

2. The woman can row at a rate of 5mph.
To calculate the time taken to row across the lake, we divide the distance by the rate: Time = Distance / Rate.
Therefore, the time taken to row across the lake is 8 / 5 or 1.6 hours.

To find the shortest amount of time, we need to compare the time taken to walk along the shore (4π / 5 hours) with the time taken to row across the lake (1.6 hours).

Since 1.6 is less than (4π / 5), it would take the woman a shorter time to row across the lake rather than walk along the shore. Therefore, the shortest amount of time it would take her to reach point C is 1.6 hours.

To find the shortest amount of time for the woman to reach point C, we need to compare the time it would take for her to walk along the shore to the time it would take for her to row across the lake.

Let's start by finding the time it would take for her to walk along the shore. The distance she needs to cover is equal to the circumference of the circle that represents the lake. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.

In this case, the radius is given as r=4, so the circumference of the lake is C = 2π(4) = 8π.

To find the time it would take for her to walk, we can use the formula Time = Distance ÷ Speed. The distance she needs to cover is 8π, and her walking speed is 10 mph.

Time to walk = (8π) ÷ 10.

Now let's find the time it would take for her to row across the lake. Since point C is diametrically opposite point A, it means that the distance she needs to row is equal to the diameter of the lake, which is twice the radius.

The distance to row = 2r = 2(4) = 8.

Using the formula Time = Distance ÷ Speed, the time it would take for her to row is:

Time to row = 8 ÷ 5.

Now, compare the two times (Time to walk and Time to row) and determine which one is shorter. This will give you the shortest amount of time it would take for her to reach point C.

I drew a circle and placed the woman at point A.

She wants to go to point B, clearly where AB is a diameter. Label the centre as O.
Pick a point P on the circle so that she will row from A to P, and then walk along the circumference from P to B
I labeled AP = x and arc PB as a (a for arc)

let the central angle for arc a be Ø radians
so a = 4Ø (a formula you should know)
so the time to walk along the arc will be 4Ø/10 or 2Ø/5

now if angle POB = Ø, then angle POA = π - Ø
recall that cos(π-Ø) = cosπcosØ + sinπsinØ
= (-1)cosØ + (0)sinØ
= -cosØ

Now in triangle PAO, using the cosine law:
x^2 = 4^2 + 4^2 - 2(4)(4)cos(π-Ø)
= 32 - 32(-cosØ) , from above
= 32 + 32cosØ
x = √(32 + 32cosØ) = (32+32cosØ)^(1/2
and the time rowing = (32+32cosØ)^(1/2)

total time = T
= (32+32cosØ)^(1/2) + 2Ø/5

d(T)/dØ = (1/2)(32+32cosØ)(-1/2) sinØ + 2/5
= sinØ/√2(32+32cosØ) + 2/5
=0 for a min of T

sinØ/2√(32+32cosØ) = -2/5
square both sides
sin^2 Ø/(4(32+32cosØ)) = 4/25
(1- cos^2 Ø)/(32+32cosØ) = 16/625
512+515cosØ = 625 - 625cos^2 Ø
625cos^2 Ø + 512cosØ - 113 = 0

wow, this factors!
(625cosØ - 113)(cosØ + 1) = 0

cosØ = 113/625 or cosØ = -1
Ø = 1.3889965 or Ø = π or 180° , which is not feasible

plug that into T
min time = √(32 + 32(113/625) + 2(1.3889965)/5
= appr 6.7 hrs

WOW#2, nice question, but you better check all that messy arithmetic.

As usual, draw a diagram. If the angle subtended by the arc walked around the lake is θ, then the distance traveled

on foot = 4θ
by boat = √(32(1+cosθ)) = 8cos(θ/2)
Use the law of cosines to get this.

So, the total time t is

t = 4θ/10 + 8/5 cos(θ/2)
dt/dθ = 2/5 - 4/5 sin(θ/2)
= 2/5 (1-2sin(θ/2))
dt/tθ = 0 when θ = π/3 or 2π/3

I'll leave it to you to figure out which is the min or max. Better check my math while you're at it.