Solving a Tricky Calculus Equation: 24 = 10(ex/20 + e-x/20)

[Larsani]

I'm reviewing for calculus and I've been stuck on this problem for a couple days. Can't seem to figure it out. I keep coming up with ln2.4 = x - x which is just silly, or x = √(400ln2.4) ≈ 18.7. I know the answer is x ≈ 12.5. But I can't figure out the correct process.

Homework Statement

24 = 10(e^x/20 + e^-x/20)

Homework Equations

The Attempt at a Solution

2.4 = e^x/20 + e^-x/20

ln2.4 ≠ x/20 - x/20

I tried using e^x/20 = e^x - e²⁰ but I don't think this is right either. Came up with

20.69 - ln e^-x = x

and this is just another silly answer. I must be forgetting some rule. It seems to me like the variables keep canceling out, which I know can't be right.

 

[daveb]

Do you know the representations for the hyperbolic functions?

 

[Larsani]

I guess not.

 

[Mark44]

I'm reviewing for calculus and I've been stuck on this problem for a couple days. Can't seem to figure it out. I keep coming up with ln2.4 = x - x which is just silly, or x = √(400ln2.4) ≈ 18.7. I know the answer is x ≈ 12.5. But I can't figure out the correct process.

Homework Statement

24 = 10(e^x/20 + e^-x/20)

Homework Equations

The Attempt at a Solution

2.4 = e^x/20 + e^-x/20

ln2.4 ≠ x/20 - x/20

The step above is incorrect. There is NO property of logs that allows you to simplify log(A + B) to logA + logB.

Multiply all three terms in the original equation by e^x/20. That will give you an equation that is quadratic in form. With a suitable substitution, you can use the Quadratic Formula to solve that equation.

I tried using e^x/20 = e^x - e²⁰ but I don't think this is right either. Came up with

20.69 - ln e^-x = x

and this is just another silly answer. I must be forgetting some rule. It seems to me like the variables keep canceling out, which I know can't be right.

 

[Villyer]

To expand on Mark's idea, if e^x/20 = u, then what does the equation change into?

 

[Larsani]

Thanks Mark! I had tried multiplying by e^x/20 and using the quadratic forumula but I guess I didn't do it right the first time. Checked the trash can: I made a calculator error and didnt notice it. Hate it when I do something right and then make a little mistake that makes me think I am not doing it right and I end up trying things that I know look wrong. Thanks again.