Optimize Rower's Trip: Find Min. Time From P to R

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In summary, the rower needs to find the optimal distance from point Q to land the boat in order to minimize the total time for the trip PR. The correct equation for this is t = 2(x^2+9)^(1/2) + 4(5-x). However, the derivative should be 4x/(x^2+9)^(1/2) - 4, not 2x/(x^2+9)^(1/2) - 4. This results in a solution of x = 3.75 km, or 3.75 km away from point Q.
  • #1

Homework Statement

A rower in a boat at a point P, 3km from the closest point Q on a straight shorline, wishes to reach a point r which is 5 km along the shoreline from Q. If he can row at 2km/h and walk at 4km/h along the shoreline, how far from point Q should the rower land the boat in order that the total time for the trip PR is minimized.

The Attempt at a Solution

Okay so if I let x be the distance from point Q, then the equation should be:

t = 2(x^2+9)^(1/2) + 4(5-x)



dt/dx = 2x/(x^2+9)^(1/2) - 4
0 = 2x/(x^2+9)^(1/2) - 4
4(x^2+9)^(1/2) = 2x
(x^2+9)^(1/2) = 1/2x
x^2+9 = 1/4x^2

but that gives no solution.

Could someone point me to where I went wrong please?
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  • #2
Correct, the distance to row is sqrt(x^2+9) and the distance to walk is 5-x

You got the times wrong. Speed = distance / time, not distance * time.
  • #3
"Let x be the distance from Q to the point on the shoreline at which the boat lands" is much more infomative than "let x be the distance from point Q"!

1. How does optimizing a rower's trip save time?

Optimizing a rower's trip involves finding the most efficient route from point P to point R, which minimizes the time it takes to travel between the two points. By taking the most direct and efficient path, the rower can save time and energy during their journey.

2. What factors are considered when optimizing a rower's trip?

The main factors considered when optimizing a rower's trip include the distance between points P and R, the current and potential wind and water conditions, the rower's skill level and stamina, and any potential obstacles or hazards along the route.

3. How is technology used to optimize a rower's trip?

Technology such as GPS and weather forecasting tools can be used to track and predict conditions along the route, allowing for more accurate and efficient planning. Additionally, specialized rowing equipment and boats can also be designed and optimized for speed and efficiency.

4. Can optimizing a rower's trip also improve their overall performance?

Yes, optimizing a rower's trip can not only save time, but it can also improve their overall performance. By taking the most efficient route, the rower can conserve energy and reduce fatigue, allowing them to perform better and potentially achieve faster times.

5. Are there any potential drawbacks to optimizing a rower's trip?

One potential drawback of optimizing a rower's trip is that it may not account for unexpected changes in conditions or obstacles along the route. It is important for rowers to always be prepared for any potential challenges and to be flexible in their approach to their journey.

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