Answer to Question #274847 in Calculus for sumu

Question

The marketing department of Spager Ltd estimated that if the selling price of product is set at $15 per unit then the sales will be 50 units per week, while, if the selling price is set at $20 per unit, the sales will be 30 units per week. Assume that the graph of this function is linear. The production department estimates that the variable cost will be $5 per unit and that the fixed cost will be $50 per week, and special cost are estimated as $0.125x2, where x is the quantity of output. All production is sold.

(a) Show that the relationship between price (Pr) and quantity sold (x) , are given by the equation Pr = 27.5 - 0.25x.

(b) Find the revenue function, R.

(c) Find the total cost function (C).

(d) Advise the company on production and pricing policy if it wishes to maximize profits, and find the maximum profit.

Answer 1

(a) Assume that the graph of this function is linear

$$Pr(x)=mx+b$$ $$Pr(50)=m(50)+b=15$$ $$Pr(30)=m(30)+b=20$$ $$50m-30m=15-20=>m=-0.25$$ $$-0.25(30)+b=20=>b=27.5$$

The relationship between price $(Pr)$ and quantity sold $(x)$ is

$$Pr=27.5-0.25x$$

(b)

$$R(x)=xPr(x)$$ $$R(x)=x(27.5-0.25x)$$ $$R(x)=27.5x-0.25x^2$$

(c)

$$C(x)=5x+50+0.125x^2$$

(d)

$$P(x)=R(x)-C(x), x\geq 0$$ $$P(x)=27.5x-0.25x^2-(5x+50+0.125x^2)$$ $$P(x)=22.5x-0.375x^2-50$$ $$P'(x)=22.5-0.75x$$

Find the critical number(s)

$$P'(x)=0=>22.5-0.75x=0$$ $$x=30$$

If $0<x<30, P'(x)>0, P(x)$ increases.

If $x>30, P'(x)<0, P(x)$ decreases.

The function $P(x)$ has a local maximum at $x=30.$

Since the function $P(x)$ has the only extremum, the function $P(x)$ has the absolute maximum for $x\geq 0$ at $x=30.$

$$P(30)=22.5(30)-0.375(30)^2-50=287.50$$

The maximum profit is $287.50 and is obtained at $x=30$ units per week.