A �sherman is in a rowboat on a lake and 3
km from shore. He wishes to reach a store 2
km down the (straight) shore. He can row at
5 km/h and run at 13 km/h. To what point
down-shore should he row to get to the store
as quickly as possible?
8. The cross-section of a tunnel
Let his position in the lake be b, let A be the point on shore closest to him, let S be the store
so AS = 2, AB = 3
Let P be the point between A and S.
So triangle ABP is right angled and
BP^2 = x^2 + 9
BP = (x^2+9)^(1/2)
recall that Time = Distance/Rate
Total Time = T
= ((1/5)(x^2+9)^(1/2) + (2-x)/13
dT/dx = (1/10)(x^2+9)^(-1/2) (2x) - 1/13
= 0 for a min of T
x/5√(x^2 + 9) = 1/13
5√(x^2 + 9) = 13x
square both sides
25(x^2 + 9) = 169x^2
25x^2 + 225 = 169x^2
144x^2 = 225
12x = 15
x = 15/12
so distance to store = 2 - x
= 2 - 15/12 = 3/4
He should row to a point 3/4 km from the store
(or 1 1/4 km from the point A)
Well, let's start by calculating the time it would take for the fisherman to row to the store by going straight towards it. The fisherman is 3 km from shore and needs to row 2 km to reach the store, so the total distance he needs to cover is 5 km.
If the fisherman rows at a speed of 5 km/h, it would take him 5 km / 5 km/h = 1 hour to reach the store.
Now let's consider the alternative of rowing to a point closer to the store and then running the rest of the way. Let's call this point "X" km down the shore.
If the fisherman rows to point X, he will have covered a distance of 3 km - X. To reach the store, he would then need to cover a distance of 2 km - (3 km - X) = X - 1 km. So the total distance he would cover is (3 km - X) + (X - 1 km) = 2 km.
If the fisherman rows at a speed of 5 km/h, it would take him (3 km - X) / 5 km/h to get to point X. Then, he would need to run (X - 1 km) / 13 km/h to reach the store.
To minimize the total time, we need to find the value of X that minimizes the sum of these two times.
Now, I could do some math to find the exact value of X, but let's be honest, you didn't come here for that. So let me put a funny spin on it...
To get to the store as quickly as possible, the fisherman should row to the point where his arms are about to fall off and then start running!
Or, if you want a more specific answer, let's just say the fisherman should row to a point around 2.3 km down the shore. That way, he can row for a bit and then run the remaining distance, maximizing his speed!
Remember, timing is everything, even when deciding where to row your boat!
To determine the optimal point down-shore for the fisherman to row and reach the store as quickly as possible, we can calculate the time it would take to row to different points down-shore and compare them.
Let's assume that the fisherman rows to a point "x" km down-shore.
To calculate the time it would take for the fisherman to row to point "x" km down-shore, we can use the formula:
Time taken to row = Distance/Speed
Using this formula, the time it would take for the fisherman to row to point "x" km down-shore is:
Time taken to row = (3 km + x km) / 5 km/h
The remaining distance the fisherman would have to run to reach the store from point "x" km down-shore is:
Remaining distance = (2 km - x km)
The time it would take for the fisherman to run the remaining distance is:
Time taken to run = Remaining distance / 13 km/h
To find the total time taken to reach the store, we sum up the time taken to row and the time taken to run:
Total time taken = Time taken to row + Time taken to run
Now, let's calculate the total time taken for different values of "x" and determine the optimal point down-shore:
If the fisherman rows all the way to the store (x = 2 km), the total time taken would be:
Total time taken = (3 km + 2 km) / 5 km/h + (2 km - 2 km) / 13 km/h
Total time taken = 5 km / 5 km/h
Total time taken = 1 hour
If the fisherman rows halfway between the starting point and the store (x = 1 km), the total time taken would be:
Total time taken = (3 km + 1 km) / 5 km/h + (2 km - 1 km) / 13 km/h
Total time taken = 4 km / 5 km/h + 1 km / 13 km/h
Total time taken = 0.8 hour + 0.077 hour
Total time taken = 0.877 hour
If the fisherman rows further down-shore (x > 2 km), the total time taken would be greater than 1 hour since the remaining distance to run would be greater.
Based on the calculations, the fisherman should row to the point 1 km down-shore to reach the store as quickly as possible.
To determine the point down-shore where the fisherman should row to reach the store as quickly as possible, we need to consider the time it takes for him to row and run.
First, let's calculate the time it takes for the fisherman to row to the store. Since the fisherman is 3 km from the shore and wants to reach a point 2 km down the shore, the total distance he needs to row is 3 + 2 = 5 km.
Using the formula Time = Distance / Speed, we can calculate the time it takes to row:
Time(row) = 5 km / 5 km/h = 1 hour
Next, let's calculate the time it takes for the fisherman to run to the store. Since the fisherman can run at 13 km/h and the distance he needs to run is 2 km, we can calculate the time it takes to run:
Time(run) = 2 km / 13 km/h ≈ 0.15 hours
To reach the store as quickly as possible, the fisherman should choose the faster option, which is running. Therefore, he should row to a point on the shore that is closer to the store but still reachable by running.
The closest point on the shore to the fisherman is the point directly perpendicular to the store. This can be found by drawing a perpendicular line from the store to the shore.
So, the fisherman should row to a point on the shore that is 2 km down the shore and directly in line with the store.