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In my pre-calculus course lesson, I have this word problem:

Amy's electric bill can be represented by the piecewise function:
$$\begin{cases} 8.25+0.0705x, &x≤400 \\ 36.45+0.0605x, &x>400 \end{cases}$$

where $x$ is the number of kilowatt hours used. Use this function to determine the cost of her electric bill if she used $825$ kilowatt hours. Round your answer to the nearest cent.

If I calculate the first line, "Amy's" bill is $\$36.45$ for the first 35 hours, as we see.
In the second line, It takes 36.45 and adds it to the reduced rate, right? Makes sense, except, that the $0.0605x$ rate is applied to the entire bill of 825 hours, not just the remaining 425 hours?

Why does the piece-wise function apply the reduced rate to the entire bill instead of just the remaining hours? Does this mean that I'm getting billed twice for some of my hours from my electric company (assuming this is a real-world accurate function)? :P

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You are right that this is a very strange function. The cost for $400$ kilowatt hours is $\$36.45$, while the cost for $401$ kilowatt hours is $\$60.71$.

Probably the author intended to write a function that does the following:

  • $\$8.25$ service charge
  • $\$0.0705$ per hour for the first $400$ kilowatt hours
  • $\$0.0605$ per hour for kilowatt hours beyond $400$

The first piece of the function is correct, and the slope of the second piece is correct. As you point out, however, you are being charged again for the first $400$ kilowatt hours, which we need to subtract off somehow.

We could fix this error by solving $$ b + 0.0605 \cdot 400 = 36.45, $$ since any reasonable function should give us the same cost at $400$ hours in both pieces. We get $b = 12.25$, which means the second piece of the function should have been $$ 12.25 + 0.0605x \text{ if } x > 400. $$

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  • $\begingroup$ Hence why I was confused - I got the problem wrong in the lesson, because I didn't think one would be billed twice for 400 hours. Thanks for the thoughts! $\endgroup$
    – studiohack
    Jun 30, 2013 at 6:23

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