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

[Darkmisc]

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

 

[andrewkirk]

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 $c$.

 

[Delta2]

A small correction to @andrewkirk post ,the first point of intersection is $(2,f(2))=(2,\frac{e-1}{2})$.

 

[Delta2]

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).

 

[Darkmisc]

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).