Question

A manufacturing plant has a capacity of $25$ articles per week. Experience has shown that $x$ articles per week can be sold at a price of Rs $p$ each where $p=110-2x$ and the cost (in Rs) of producing $x$ articles is $600+10x+x^2$ .

The profit function represents a

• A. straight line
• B. a parabola with vertex at $(20,\frac{700}{3})$
• C. a parabola with vertex at $(\frac{50}{3},\frac{700}{3})$ and focus $(\frac{50}{3},\frac{2199}{3})$
• D. a parabola with vertex at $(\frac{50}{3},\frac{700}{3})$ and focus $(\frac{50}{3},\frac{2099}{3})$

Answer 1

The correct option is D a parabola with vertex at $(\frac{50}{3},\frac{700}{3})$ and focus $(\frac{50}{3},\frac{2099}{3})$

$P=$ Revenue - cost

$=xp-(600+10x+x^2)=x(110-2x)-(600+10x+x^2)=100x-600-{3x}^2$

$=-3[{(x-\frac{50}{3})}^2-\frac{700}{9}]$

So, $P-\frac{700}{3}=-3{(x-\frac{50}{3})}^2$

This represent a parabola with vertex at $(\frac{50}{3},\frac{700}{3})$ and focus $(\frac{50}{3},\frac{2099}{12})$.