Review practice · Section 2 of 2

The graph of $y=f(x)$, where $f(x)$ is a cubic polynomial, passes through the points $(1,6)$, $(-2,15)$ and has two stationary points at $x=1/3$ and $x=-2$.

Find the equation of the curve and hence find its $x$-intercept.5 marks

A cubic polynomial $p(x)$ leaves a remainder of 52 when divided by $x-2$. Given that the curve with equation $y=p(x)$ passes through the point $(1.5,32.375)$ and has a stationary point at $(-1,-2)$, find $p(x)$.4 marks

A brick manufacturer has a depot with a stock of 30 000 bricks. He receives orders from three builders P, Q and R for a total of 30 000 bricks and the total order from builders P and R is twice that of builder Q.

(a) If the profit earned from each brick is $3 and the manufacturer earns $9 600 more in profit from builder R compared to builder P, find the number of orders from each builder.4 marks

(b) Instead of using the information in part (a), it is now given that the cost of transporting each brick to builders P, Q and R from the depot is $0.04, $0.02 and $0.03 respectively. If there is a minimum order of 1000 bricks from each builder, what are the orders that the manufacturer should accept so as to keep the cost of transportation a minimum?3 marks

To add to its existing inventory of shoes, a footwear company plans to manufacture 20 pairs of sports shoes with a budget of $980. The company manufactures three types of sports shoes: Walkers, Runners and Trainers, whose manufacturing costs per pair are $30, $40 and $50 respectively. Find the possible combinations of the number of pairs of each type of sports shoes that the company can manufacture.4 marks

If the company plans to sell each pair of Walkers, Runners and Trainers at $80, $130 and $150 respectively, how many pairs of each type of sports shoes should the company manufacture in order to maximise profit?2 marks