how to slove it

Question

if the price is given p(q)=49-q at q units sold and the total cost is given c(q)=1/8q^2+4q+200 for producting the q units. Determine the level of production qu where the profit p(q) is maximized.

Answer 1

If I am understanding this correctly:

Units are sold at a price of 49-q for q units sold:

This means Revenue = q(49-q) = -q² + 49q

 

Cost for making the q units is:

(1/8)q² + 4q + 200

 

Profit = Revenue - Cost

Profit = -q² + 49q - ((1/8)q² + 4q + 200)

        = -q² + 49q - (1/8)q² - 4q - 200

        = (-9/8)q² + 45q - 200

 

The profit function graphs as a parabola opening downward.

The maximum profit will be at the vertex of the parabola.

The q coordinate of the vertex is -b/2a from the quadratic equation.

a = -9/8, b = 45

 

-b/2a = -45/[2(-9/8)] = -45/(-9/4) = 45(4/9) = 20 units

 

The profit for 20 units is:

(-9/8)(20)² + 45(20) - 200

= -450 + 900 - 200

= 250

 

The maximum profit of 250 is obtained by the production/sale of 20 units

***************************************

Check the functions determined at the beginning given q = 20:

 

Price: 49-q = 29

 

Revenue: -q² + 49q = -400 + 980 = 580

 

Cost: (1/8)q² + 4q + 200 = 50 + 80 + 200 = 330

 

Profit: 580 - 330 = 250