Function as mathematical model problem.?

Question

given are the respective price demand equation and cost functions for the sale of x thousand watches

p= 75- 3x

c(x)= 150 + 16x

where the cost is given in thousands of pesos

a.) find a mathematical model expressing revnue as a function of x

b.) what quantity maximizes revenues?

c.) mathematical model expressing the profit as a function of x

d.) find the break even points

e) for what outputs will a profit occur?

f.) what quantity will maximize profits?

please show complete solution for each question. thank you very much!!!!!

Answer 1

a)

Revenue = price multiplied by number sold (x)

R = px = 75x - 3x^2

b)

Find the vertex of the above parabola.

x = -b/(2a) = -75/(2*(-3)) = 75/6 = 12.5  (12500 watches)

c)

P = R - c

P(x) = 75x - 3x^2 - (150 + 16x) = -3x^2 + 59x - 150

d)

Where the profit is first = 0 is the break-even point.

    x = 3  or x = 16.666...    (x = 3 is the break-even point)

3000 watches must be sold to break even.

e)

Anwhere that P(x) is positive.

between x = 3 and 16.666...

Or between 3000 and 16,667 watches.

f) Find the vertex of the parabola...

x = -b/(2a) = -59 / (2(-3)) = 59/6 = 9.8333...

Or 9,833 watches will maximize profits.

Hope that helps.