Given the exercise in the screenshot below, I don't understand why, in order to find the value of the constant 'r', we need to equate r2 to 0.55 (as they did in the screenshot), when we actually need to equate the whole function, which also includes the 'c' variable.
I could understand why we would equate cr2=0.55 and then solve it, because the c variable is also part of the function, and if we are given that after 2 minutes it has 55% left, then we have to include what is left from, and that "from" has to include the entire amount, which must include the "c" variable, which is basically the initial amount.
Then why do they solve the equation as shown in the hints, just with r2? How is that logical, and why using the "c" variable is illogical, if I'm wrong. Thanks!
Screenshot: https://i.stack.imgur.com/AUAR9.png
To see why $c$ can be ignored consider the following:
Since we know that, after 2 minutes, the amount of sugar in the gum is 45% less than it was, we know that gum now has 55% of the sugar it had 2 minutes ago. In equation form, if $c\cdot r^t$ is the amount of sugar before those two minutes transpired, we have
$0.55c\cdot r^t=cr^{t+2}$
You can immediately divide both sides by $c\cdot r^t$ to obtain
$0.55=r^2$.
Solve for $r$ (and take the positive solution). Now you just need to find $c$. To do that, use the fact that, at $t=5$ you know that $c=3$ grams (given in problem). This amounts to solving
$3=c\cdot r^5$
for $c$ where $r$ is now a known value (because of our previous work).