Shimaera,
Profit, P(x), is revenue minus cost and revenue is the price you charge per item, time the number of items you sell, thus
P(x) = revenue  cost
= p(x) x  C(x)
= x (
250+2x/10)  (0.015x^{2}+12x+1000)
Simplify this expression and then the profit when 500 items are produced is P(500).
The marginal profit is the additinal profit obtained by producing one more item. This is approximated by the derivative of the profit with rrespect to the number of items produced. Hence find the derivative of P(x) and evaluate it at x = 500.
For question 2 let t be the time after 9 am in hours. Since the car is traveling 100 km every hour, after t hours it has progressed 100 t km north of Marytown. The truck has traveled 70 t km east of Marytown.
Use Pythagoras Theorem to find the distance between the car and the truck (the length of the red line segment in the diagram) and then differentiate this expression to find the rate at which the distance between them is changing.
Penny
