| Date | May Specimen | Marks available | 2 | Reference code | SPM.1.sl.TZ0.15 |
| Level | SL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find | Question number | 15 | Adapted from | N/A |
Question
A small manufacturing company makes and sells \(x\) machines each month. The monthly cost \(C\) , in dollars, of making \(x\) machines is given by
\[C(x) = 2600 + 0.4{x^2}{\text{.}}\]The monthly income \(I\) , in dollars, obtained by selling \(x\) machines is given by
\[I(x) = 150x - 0.6{x^2}{\text{.}}\]\(P(x)\) is the monthly profit obtained by selling \(x\) machines.
Find \(P(x)\) .
Find the number of machines that should be made and sold each month to maximize \(P(x)\) .
Use your answer to part (b) to find the selling price of each machine in order to maximize \(P(x)\) .
Markscheme
\(P(x) = I(x) - C(x)\) (M1)
\( = - {x^2} + 150x - 2600\) (A1) (C2)
\( - 2x + 150 = 0\) (M1)
Note: Award (M1) for setting \(P'(x) = 0\) .
OR
Award (M1) for sketch of \(P(x)\) and maximum point identified. (M1)
\(x = 75\) (A1)(ft) (C2)
Note: Follow through from their answer to part (a).
\(\frac{{7875}}{{75}}\) (M1)
Note: Award (M1) for \(7875\) seen.
\( = 105\) (A1)(ft) (C2)
Note: Follow through from their answer to part (b).
