need help solving 0.687t^1.25 + 1.67t = 45

[LouV]

Need help in solving following equation:

0.687t^1.25 + 1.67t = 45

 

[pka]

I would suggest graphing the following:
\(\displaystyle \L0.687t^{1.25} + 1.67t - 45\).

I found an approx. root of 14.901705725399374276.

 

[LouV]

Thank you for your respose PKA. I was hoping to find another way of solving this equation instead of graphing. I am just not sure how one would set it up for solution.

 

[pka]

I doubt that there is what we could call a ‘closed form’ solution for this. Because \(\displaystyle t^{1.25} = t^{\left( {\frac{5}{4}} \right)} = \sqrt[4]{{t^5 }} = t\sqrt[4]{t}\), there is on obvious substitution to simplify the equation.

 

[galactus]

You could try the Intermediate Value Theorem to hone in on the solution.

If t=15, then you have 0.330133...

If t=14, then -3.015552....

See the sign change?. That means the solution is somewhere between 14 and 15.

Keep narrowing it down until you get as close as you want.

 

[Denis]

...and lots of ways to narrow down; mine would be:
t = 45000 / (687t^(1/4) + 1670)

 

[LouV]

Yes. I can see now that no closed form solution exists and iteration is the way to go. I was afraid I was missing something obvious.
Thank you all for pointing me in the right direction.