Manufacturing profit curve

Optional extension

A manufacturer’s total profit, in dollars, from \(x\) toys is \(P=30x-180-\dfrac{x^2}{4}\), where \(0\le x\le100\).

The curve is a continuous mathematical model; actual production must be a whole number of toys. For the target profit, give the continuous-model estimates and decide whether an integer production level attains it exactly.

1. Find the missing table values \(p,q\).[2]
2. Plot the graph.[2]
3. Use it or calculation to find
   1. the profit for 45 toys,[2]
   2. the production levels giving \(\$650\) profit,[2]
   3. the production level and total profit at the maximum, and the corresponding profit per toy, and[2]
   4. the profit for 6 toys and its meaning.[2]

For graph readings, give values to the nearest 0.1 unit where an exact value cannot be read from the grid. Use the labelled axes and grid provided.