## anonymous 4 years ago 49t+245e^(-t/5)-545=0, by some other means than a calculator or wolfram?

1. ash2326

if x is very small than 1, then e^x can be expanded as 1+x+x^2/2! and higher order terms can be neglected let's assume -t/5 << 1 then e^(-t/5)=1-t/5+t^2/50 substitute this in the equation 49t+245(1-t/5+t^2/50)-545=0 49t+245-49t+4.9t^2-545=0 combining like terms 4.9t^2=300 $t=\sqrt{\frac{300}{4.9}}$ $t=7.8246$

2. anonymous

cool man, series expansion, thanks

3. phi

but t/5 is not << 1

4. ash2326

phi we don't know about that, so I just assumed it

5. anonymous

it would be if the premise is that t is small to begin with, in that case t/5 would be smaller

6. anonymous

but it is that assumption

7. anonymous

thanks unless you have a belligerent way to do it phi

8. phi

9. ash2326

phi you're right it won't work, we'll have to include higher order terms

10. anonymous

not yet, I just wanted to get an idea of where to go before I worked on it. It seems reasonable if I do an longer expansion

11. anonymous

I just didn't think of doing the expansion in the first place

12. anonymous

rather couldn't

13. anonymous

I've got to roll, but thanks for the help guys

14. ash2326

Yeah daomowon , now you include the cubic power, find t and substitute back, if it doesn't then include higher power

15. mathmate

Do you mean no computers and no wolfram? A problem like this without calculators would take a while!

16. phi

It's not clear what the constraints are on how to do this problem. But generally, you would solve it using numerical techniques. such as http://en.wikipedia.org/wiki/Newton's_method see attached pdf for result

17. mathmate

I would approach this using numerical methods, like Newton's method. We should note that the middle (e^(-t/5) term is insignificant when t>0, which gives the first approximation as x=545/49=11.1. It converge to10.5116 in a few more iterations. When t<0, both terms come into play, but e^(-t/5) has more influence. It should do the same towards -6.

18. anonymous

yeah, I figure if I can get at these types of things without computers and stuff, I can get at wider selection of material better in the long run. thanks for all the insight though, I couldn't make the leap I needed in the beginning, but this all helps a whole lot