In Argentine, the cost of kW/h is $4,5$ cents and in Latin America, $16$ ctvs (average).
a) In how many months would you update the cost to reach $16$ cents kW/h, applying a linear $20\%$ increment each month? And with $40\%$?
RE: I came to the conclusion that it will cost $12,77$ months with $20\%$ and $6,39$ months for $40\%$. And that with an increment of $255\%$ in $1$ month I would reach it.
Now, the next point is the one I am having trouble with:
b) Elaborate a graphic representation that allows to visualize and analyze how the quantity of months needed varies according to the monthly increment percentage.
I think this is a linear equation, right? Well I am having trouble to reach the $y = m x + b$ equation. What should I use?
c) And the minimun amount of months is $1$, so should I limit the function domain, like saying $y > 1$ ?
Thanks!!!
For part a)
Let us use numbers and call $P_0$ the price before any increase. After a month, the price is increased by $x$, so at the start of next month, the price will be $$P_1=(1+x)P_0$$. Similarly, at the start of next month, the price will be $$P_2=(1+x)P_1=(1+x)^2 P_0$$ and so on. I suppose you see how this works and then, at the start of month $n$, the price will be $$P_n=(1+x)^nP_0$$ So, if $P_0=4.5$ and you want to use $x=0.20$ to reach $P_n=16$, you then need to solve for $n$ $$16=(1+0.2)^n \times 4.5$$ Taking the logarithms of both sides, you then end with $n=6.96$ that is to say $7$ months.
If we repeat the same calculation with $x=0.4$, you need to solve similarly $$16=(1+0.4)^n \times 4.5$$ whih leads to $n=3.77$ that is to say $4$ months.
For part b)
In part a), you saw that basically the equation can write $$\frac{16}{4.5}=(1+x)^n$$ so, taking again logarithms, we have $$n \log(1+x)=\log(\frac{16}{4.5})$$ and since tou want $n$ as a function of $x$, you have $$n=\frac{\log(\frac{16}{4.5})}{\log(1+x)}$$ If you want to do all of that in a single month, use $n=1$ in the previous formula and, using exponentials, solve for $x$ and you will find $x=2.55556$ that is to say a $255.56$% increase which corresponds exactly to the ratio of $\frac{16-4.5}{4.5}$
I hope that you percieve why we do not face linear relations.