A certain paperback sells for $12. The author is paid royalties of 11% on the first 10,000 copies sold, 12.5% on the next 5000 copies, and 18% on any additional copies. Find a piecewise-defined function R that specifies the total royalties if x copies are sold.
Please help me figure out the rest of this equation. Here is what I have so far.
R(x)=
_____if 0≤x≤ 10000
_____if 10000 < x ≤ 15000
_____if x > 15000
Any help is appreciated! Thanks!
if 0≤x≤ 10000 : R(x) = $12 * 11% *x
at x = 10000, R(x) = $12 * 11% *10000 = 13200
10000 < x ≤ 15000 : R(x) = 13200 + $12*12.5%*(x-10000)
(The profit made on the first 10000 books + the profit made on books sold above 10000)
at x = 15000, R(15000) = 13200 + $12*12.5%*(15000-10000) = 20700
x > 15000
R(x) = 20700 + 12*.18*(x-15000),
(The profit made on the first 15000 books + the profit made on books sold above 15000)
R(x)= 0.11x if 0 ≤ x ≤ 10000
0.11(10000) + 0.125(x-10000) if 10000 < x ≤ 15000
0.11(10000) + 0.125(15000-10000) + 0.18(x-15000) if x > 15000
To find the piecewise-defined function R that specifies the total royalties, we need to calculate the royalties for each range of copies sold.
Let's break it down step-by-step:
1. For 0 ≤ x ≤ 10,000 copies sold:
The author is paid royalties of 11%. So, the royalties for this range can be calculated as:
Royalties = 0.11 * x
2. For 10,000 < x ≤ 15,000 copies sold:
The author is paid royalties of 12.5% for the next 5,000 copies. So, the royalties for this range can be calculated as:
Royalties = (0.11 * 10,000) + (0.125 * (x - 10,000))
3. For x > 15,000 copies sold:
The author is paid royalties of 18% for any additional copies beyond 15,000. So, the royalties for this range can be calculated as:
Royalties = (0.11 * 10,000) + (0.125 * 5,000) + (0.18 * (x - 15,000))
Putting it all together, the piecewise-defined function R(x) for total royalties is:
R(x) =
0.11x if 0 ≤ x ≤ 10,000
0.11(10,000) + 0.125(x - 10,000) if 10,000 < x ≤ 15,000
0.11(10,000) + 0.125(5,000) + 0.18(x - 15,000) if x > 15,000
I hope this helps! Let me know if you have any further questions.
To determine the different intervals and corresponding royalties in the piecewise-defined function, we need to break down the given information.
First, we know that the author is paid royalties of 11% on the first 10,000 copies sold. So, within the range of 0 to 10,000 copies, the royalties would be 11% of the price of each copy, which is $12.
Next, the author receives 12.5% royalties on the next 5,000 copies sold. This means that within the range of 10,001 to 15,000 copies, the royalties would be 12.5% of the price of each copy, which is $12.
Finally, for any additional copies sold beyond 15,000, the author receives 18% royalties. This means that for x copies greater than 15,000, the royalties would be 18% of the price of each copy, which is $12.
Now, let's fill in the missing parts of the piecewise-defined function:
R(x) = 0.11x, if 0 ≤ x ≤ 10,000
R(x) = 0.125x, if 10,001 < x ≤ 15,000
R(x) = 0.18x, if x > 15,000
This function specifies the total royalties, R, based on the number of copies sold, x.