anonymous one year ago The variables x and y satisfy the equation (x)^ny=C, where n and C are constants. When x=1.10, y=5.20, and when x=3.20, y=1.05. (i) Find the values of n and C.

1. anonymous

$x ^{n}y=C$

2. anonymous

@Michele_Laino

3. Michele_Laino

if we substitute your data, we get two different conditions, namely: $\Large \begin{gathered} {\left( {1.1} \right)^n} \cdot 5.2 = C \hfill \\ \hfill \\ {\left( {3.2} \right)^n} \cdot 1.05 = C \hfill \\ \end{gathered}$ those equation are an algebraic system, which can be solved for n and C

4. Michele_Laino

equations*

5. anonymous

not being able to solve -_-

6. Michele_Laino

if I use the elimination method, I can write this: $\Large {\left( {1.1} \right)^n} \cdot 5.2 = {\left( {3.2} \right)^n} \cdot 1.05$

7. anonymous

yesss

8. Michele_Laino

now, I divide both sides of that equation by (1.1)^n, so I get: $\Large 5.2 = {\left( {\frac{{3.2}}{{1.1}}} \right)^n} \cdot 1.05$

9. Michele_Laino

then I divide both sides again by 1.05, so I can write this: $\Large \frac{{5.2}}{{1.05}} = {\left( {\frac{{3.2}}{{1.1}}} \right)^n}$

10. anonymous

okay

11. Michele_Laino

we got an exponential equation, which can be solved using logarithms

12. anonymous

13. Michele_Laino

why?

14. anonymous

i dont know.. can you continue further ?

15. Michele_Laino

ok!

16. Michele_Laino

if we take the logarithm in base 10, of both sides, we get: $\Large \begin{gathered} n \cdot {\log _{10}}\left( {\frac{{3.2}}{{1.1}}} \right) = {\log _{10}}\left( {\frac{{5.2}}{{1.05}}} \right) \hfill \\ \hfill \\ n = \frac{{{{\log }_{10}}\left( {\frac{{5.2}}{{1.05}}} \right)}}{{{{\log }_{10}}\left( {\frac{{3.2}}{{1.1}}} \right)}} \hfill \\ \end{gathered}$

17. Michele_Laino

what do you get?

18. anonymous

finally got the answer.. thank you very much

19. Michele_Laino

thanks! :)