# Help with what I think is a differential problem

• perjac
In summary, the conversation is about a math problem involving a man on a kayak trying to reach his destination in the least possible time. The solution involves using the Pythagorean theorem and taking the derivative to find the minimum distance from the nearest point on the beach.
perjac
Hi All,

Did my maths B exam the other day but was absolutely stumped by one question. Spent half an hour thinking about it only to give up.

Question:

A man on a kayak (K) is 3 kilometres out to sea from the nearest point, (O) of a straight beach. His destination (D) is 6 kilometres along the beach from O. The fastest he can paddle is 4 km/hr and his maximum walking speed is 5km/h. How far from O should he go ashore to reach his destination in the least possible time?

Anyways the question was alongside a whole heap of differential problems so I assume you would need to differentiate an equation, find the min SP and that should tell you the distance but I have no idea how to get to the equation or if I was even on the right track.

This was the only question I couldn't answer and it is driving me nuts.

Thanks,

Perjac

Let P be the point at which the kayak lands on the beach. Let x be the distance from O to P. By the Pythagorean theorem, he must have paddled a distance $\sqrt{x^2+ 9}$ kilometers. At 4 km/hr, that will require $(1/4)\sqrt{x^2+9}$ hours. He then has to walk 6- x kilometers. At 5 km/hr, that will require $(1/5)(6- x)$ hours. The total time is $(1/4)\sqrt{x^2+ 9}+ (1/5)(6- x)$ hours. You want to minimize that. Take the derivative with respect to x and set it equal to 0.

## 1. What is a differential problem?

A differential problem is a mathematical problem that involves finding the rate of change of a variable or function over a given interval. It is typically solved using differential equations, which are mathematical equations that describe how a variable changes in relation to other variables.

## 2. How do I know if I have a differential problem?

If you are given a problem that involves finding the rate of change of a variable or function, then you likely have a differential problem. These types of problems often involve real-world scenarios where a quantity is changing over time or space.

## 3. What are some common techniques for solving differential problems?

The most common techniques for solving differential problems include separation of variables, substitution, and the use of specific formulas for different types of differential equations. It is important to have a strong understanding of calculus and algebra to effectively solve differential problems.

## 4. Are there any tips for solving differential problems?

One helpful tip for solving differential problems is to first identify the type of differential equation you are dealing with. This will help guide you towards the appropriate technique for solving the problem. Additionally, it is important to carefully read the problem and clearly define any given variables before attempting to solve.

## 5. What are some common applications of differential problems?

Differential problems have a wide range of applications in various fields such as physics, engineering, economics, and biology. They can be used to model and analyze real-world situations such as population growth, heat transfer, and chemical reactions. Understanding how to solve differential problems is essential for many scientific and mathematical disciplines.

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