Solving a Problem: Have I Made a Mistake or Are the Solutions Wrong?

In summary, the conversation discusses the process of finding the values of a, b, and c by solving equations and using first principles. The solutions given in the conversation lead to a value of b as 2ln5, but there is a small correction needed and the correct value is actually 2log5. This allows for the equation to be solved to find the value of c.
  • #1
Darkmisc
220
31
Homework Statement
Is there a mistake in the below solution?
Relevant Equations
Definite integrals
Hi everyone

To solve the below problem, I assumed the affected area was 2x2 minus the definite integral of the given function between 2 and 4.

I then equated the answer for that with the given function to solve for a, b and c.

I don't know why the solutions give b as 2ln5.

Have I made a mistake, or are the solutions wrong?

Thanks
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  • #2
You are trying to work out a, b and c by solving equations. That cannot work, as you have three unknowns and only one equation. Instead you set the values a and b from first principles, as the lower and upper bounds of x at which the fire front intersects the farm. Given the fire front equation is ##f(x) = \frac12 e^{\frac x2}-\frac12## and the farm is ##[2,4]\times [0,2]## we see that the intersection points are ##(2,e^\frac12)## and ##(b,2)##, the second point being where the fire front intersects the line ##y=2##. That second point gives us the equation
$$2 = f(b)=\frac12 e^\frac b2-\frac12$$
which we solve to get
$$b=2\log 5$$
Now that you know ##a## and ##b## you can solve the equation to find ##c##.
 
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  • #3
A small correction to @andrewkirk post ,the first point of intersection is ##(2,f(2))=(2,\frac{e-1}{2})##.
 
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  • #4
I also think that the "4=(4-2)x(2-0)" in your equation for A shouldn't be 4 but instead ##(b-2)\times(2-0)=2(b-2)##, hard to explain with words without making a scheme (I am really bad in making schemes).
 
  • #5
Yeah, I drew the diagram for myself wrong. I assumed the fire front would touch the right edge of the property (which it didn't).
 
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FAQ: Solving a Problem: Have I Made a Mistake or Are the Solutions Wrong?

1. How can I determine if I have made a mistake?

One way to determine if you have made a mistake is to double check your work and go through each step of the problem to see if there are any errors. You can also ask a colleague or peer to review your work and provide feedback.

2. What should I do if I realize my solution is wrong?

If you realize your solution is wrong, don't panic. Take a step back and try to identify where the mistake was made. Once you have identified the mistake, go back and correct it. If you are still struggling, don't hesitate to ask for help from a teacher or mentor.

3. Is it possible to have multiple solutions to a problem?

Yes, it is possible to have multiple solutions to a problem. Some problems may have more than one correct answer, while others may have a range of possible solutions. It is important to carefully consider and evaluate each solution to determine which one is the most appropriate for the given problem.

4. What steps can I take to avoid making mistakes in the future?

To avoid making mistakes in the future, it is important to carefully review and check your work, as well as double check any calculations or steps involved in solving the problem. You can also try to break down the problem into smaller, more manageable parts and tackle them one at a time. It may also be helpful to seek feedback and advice from others.

5. How can I improve my problem-solving skills?

Improving problem-solving skills takes practice and patience. One way to improve is to regularly engage in problem-solving activities and challenges. You can also try to approach problems from different perspectives and think outside the box. Reflecting on your problem-solving process and seeking feedback from others can also help you identify areas for improvement.

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