# Optimization -writing and solving the equations

• Poppynz
In summary, the problem involves finding the optimal landing point for a boat to reach a town 10km away in the least amount of time, given that the boat can row at 3km/h and walk at 5km/h. The equation for this is y = (1/3)*(4^2 +x^2)^(1/2) + ((10-x)/5), and the solution involves differentiating the equation and setting it equal to 0 to find the critical point. However, the previous attempt had an error in squaring the 5, resulting in imaginary solutions.
Poppynz
Hi
Im trying to write an equation for the question below. Could someone please point me in the right direction with writing it?

## Homework Statement

An island is 4km from the nearest point p on the straight shoreline of a lake. if a person can row a boat at 3km/h and walk at 5km/h where should the boat be landed to arrive at a town 10km away is the least time?

## The Attempt at a Solution

I think the equation is y=(1/3)*(4^2+x^2)^(1/2)+((10-x)/5)

this doesn't look hard to differentiate but i can't seem to get the right answer-
in fact i get imaginary numbers

this is what i did

y=(1/3)*(16+x^2)^(1/2)-(1/5*x)+2
y'= -1/5+1/2*1/3(16+x^2)^(1/2)*2x
y'= -1/5+(2x/(6(16+x^2)^(1/2))
y'=0
1/5=x/(3(16+x^2)^(1/2))
3*(16+x^2)^(1/2)=5*x //square both sides
9*(16+x^2)=5*x^2
144+9x^2-5x^2=0
x= +/- 6i //this is obviously wrong as a person does not travel imaginary distances

Welcome to PF!

Hi Poppynz! Welcome to PF!

(have a square-root: √ and try using the X2 tag just above the Reply box )
Poppynz said:
… 3*(16+x^2)^(1/2)=5*x //square both sides
9*(16+x^2)=5*x^2
144+9x^2-5x^2=0
x= +/- 6i //this is obviously wrong as a person does not travel imaginary distances

oops! … you forgot to square the 5, so you got 5 - 9 instead of 25 - 9

Try again!

haha such a silly mistake thanks :)

## 1. What is optimization in scientific writing?

Optimization in scientific writing refers to the process of finding the best or most efficient solution to a problem, typically through the use of mathematical equations and models.

## 2. How do you write equations for optimization problems?

To write equations for optimization problems, you must first identify the objective function (what you are trying to optimize) and the constraints (limitations or rules that must be followed). Then, you can use mathematical notation to express the objective function and constraints in terms of variables and parameters.

## 3. What is the difference between analytical and numerical solutions in optimization?

Analytical solutions involve solving equations algebraically to find an exact solution, while numerical solutions use computational methods to approximate a solution. In optimization, analytical solutions may only be possible for simple problems, while numerical solutions can handle more complex problems.

## 4. How do you know if you have found the optimal solution to an optimization problem?

In scientific writing, the optimal solution is typically the one that maximizes or minimizes the objective function while satisfying all constraints. This can be determined by using mathematical techniques such as differentiation and linear programming, or by using software programs specifically designed for optimization problems.

## 5. What are some common applications of optimization in science?

Optimization has many applications in science, including in fields such as engineering, physics, economics, and biology. Some common examples include optimizing the design of structures or processes, maximizing profits or efficiency, and identifying the most effective treatment for a disease.

Replies
10
Views
1K
Replies
2
Views
832
Replies
2
Views
979
Replies
7
Views
870
Replies
27
Views
876
Replies
4
Views
1K
Replies
6
Views
1K
Replies
5
Views
2K
Replies
5
Views
1K
Replies
4
Views
1K