You are standing at the edge of a slow moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground which is one mile from the point directly across the river where you start your swim. What route will take the least amount of time?

I cannot tell what your variables represent

My sketch has the distance from the point directly across his starting point to the end of his swim as x
and the distance to be walked is then 1 -x
let the length of his swim by y
in the right-angled triangle:
x^2 + 1 = y^2
y = (1+x^2)^(1/2)

time for the swim = (1/2)(x^2 + 1)^(1/2)
time for the walk = (1/3)(1-x)

Time = (1/2)(x^2 + 1)^(1/2) + (1/3)(1-x)
d(time)/dx = (1/4)(x^2 + 1)^(-1/2) (2x) - 1/3 = 0 for min time

x/( 2√(x^2 + 1) ) = 1/3
square both sides
x^2/(4x^2 + 4) = 1/9
9x^2 = 4x^2 + 4
5x^2 = 4
x = 2/√5 = .894

He should aim for a spot .894 miles down from directly across, or .106 miles from the campground

Well, it seems you're in quite the quagmire, my friend! To find the route that takes the least amount of time, we need to do a bit of calculation while keeping in mind that time waits for no one (except maybe for second-hand watches).

Let's break it down, shall we? If you choose to swim directly across the river, it will take you 0.5 hours (1 mile / 2 mph = 0.5 hours). Once you reach the opposite bank, you then have to walk 1 mile to the campground, which will take you an additional 0.33 hours (1 mile / 3 mph = 0.33 hours).

Now, if you're a bit of a strategic swimmer, I have a different clown-tastic idea for you. Instead of swimming directly across, swim diagonally at a slight angle upstream. By doing this, you'll reduce the distance you have to walk once you reach the opposite bank. The question is, at what angle should you swim?

If we do some trigonometric wizardry, we find that the optimal angle to swim at is about 57 degrees.

Why, you ask? Well, by swimming at a slight angle upstream, you'll decrease the downstream drift caused by the river's current. This means that when you finally exit the water, you'll have less distance to walk to the campground. Pretty nifty, huh?

Now, with this clever approach, you'll find that swimming diagonally and then walking from the opposite bank will take you approximately 0.43 hours (0.63 hours swimming + 0.33 hours walking).

So, my dear friend, by swimming diagonally at an angle of 57 degrees upstream, you'll zip across that river in the shortest amount of time possible. Happy swimming and even happier camping!

To determine the route that will take the least amount of time, we need to calculate the time it would take for each option and compare them. Let's break it down step-by-step:

1. Swim across the river: Since the river is one mile wide, swimming at a speed of 2 mph would take 1/2 hour (30 minutes).

2. Choose a point on the opposite bank: At this point, you have reached the opposite bank and need to choose a point to start walking. Since you want to minimize the total time, it would be best to choose a point closest to the campground.

3. Walk to the campground: At a walking speed of 3 mph, walking one mile would take 1/3 hour (20 minutes).

To minimize the total time, we need to find the point on the opposite bank that minimizes the combined swimming and walking time:

Option 1: Swim across the river, choose a point right across the campground, walk 1 mile to the campground.
Total time = 30 minutes (swimming) + 20 minutes (walking) = 50 minutes.

Option 2: Swim across the river, choose a point midway between your starting point and the campground, walk 1/2 mile to the campground.
Total time = 30 minutes (swimming) + 10 minutes (walking) = 40 minutes.

Option 3: Swim across the river, choose a point right across your starting point, walk 1 mile to the campground.
Total time = 30 minutes (swimming) + 20 minutes (walking) = 50 minutes.

Comparing the options, it is clear that Option 2 will take the least amount of time, with a total of 40 minutes. Therefore, swimming across the river and choosing a point midway between your starting point and the campground, then walking the remaining 1/2 mile to the campground, will be the fastest route.

To determine the route that will take the least amount of time, we need to compare the time it takes to swim across the river plus the time it takes to walk to the campground for different points on the opposite bank.

Let's assume you swim to a point "X" on the opposite bank. Here's how you can determine the total time for two different scenarios:

Scenario 1: Swim straight across and walk to the campground.
In this scenario, you swim directly across the river, which takes (1 mile / 2 mph) = 0.5 hours. After that, you walk 1 mile to the campground, which takes (1 mile / 3 mph) = 1/3 hours. So the total time for this scenario is 0.5 + 1/3 hours.

Scenario 2: Swim to a point closer to the campground and walk to the campground.
In this scenario, you swim to a point on the opposite bank that is closer to the campground. Let's say the distance from this point to the campground is "d" miles. The time it takes to swim to this point is (d miles / 2 mph) = d/2 hours. Then you walk the remaining distance to the campground, which is (1 - d) miles, in (1 - d) / 3 hours. So the total time for this scenario is d/2 + (1 - d) / 3 hours.

To find the point that minimizes the total time, we need to find the value of "d" that minimizes the expression d/2 + (1 - d) / 3.

To find the minimum, we can take the derivative of the expression with respect to "d" and set it to zero:

(d/2 + (1 - d) / 3)' = (1/2 - 1/3) = 1/6

Solving for "d", we get:

1/6 = 0
d = 1/6

Therefore, swimming to a point on the opposite bank that is 1/6 mile from the campground will minimize the total time. This means you should swim to a point that is approximately 0.167 miles from the campground on the opposite bank, and then walk the remaining distance to the campground.

so I know you will use

T=dr/rr +dL/rL