algebra-precalculusarithmeticpercentagesword problem
A certain car brand estimates that a certain model, that now costs
21000 euro, will devaluate 5% per year.How much will this car cost in 6 years?
I did:
$$21000 \cdot (1+.05)^6 \approx 28142$$
Now I remove the initial value from this one to get the compound devaluation
$$28142-21000 = 7142$$
Then I subtract this value from the initial value
$$21000-7142 = 13858$$
But my book says the solution is 15,437 euro.
What did I do wrong?
The current version of the function is not right. As @David implied, you do not correctly handle the fact that anything between $11$ and $14$ days will be billed the same as $2$ weeks, due to the "until the day rate exceeds the weekly rate" clause. You also do not handle $x$ being a non-integer: it should be rounded up.
Here is a function that I believe is correct:
$$f(n)=\begin{cases} 185+37(\lceil x\rceil-7), & 7\le x\le 11 \\[2 ex] 370, & 11<x\le 14 \end{cases}$$
where $\lceil x\rceil$ is the ceiling function, rounding up to the smallest integer greater than or equal to $x$.
The ceiling function means that each small section of the graph, of width $1$, will be a horizontal line. The second case means that the last part of the graph is a horizontal line between $x=11$ exclusive to $x=14$ inclusive. The first case means that the segments will be like a staircase, with the left-hand endpoints on the line $y=185+37x$. Here is a graph.
A better graph would have large solid dots and open dots to show which endpoints of the line segments are on or not on the graph. This should basically answer your question, however.
Best Answer
You're adding $5\%$ to the value each year when you multiply by $(1+.05)^6$. To subtract $5\%$ each year, use $(1-.05)^6$. It's not the same, because after the first year, you're taking $5\%$ from a smaller value each year. If it grows, you're adding $5\%$ to a larger value each year.