Calculus Max Profit

posted by .

To make pom poms in our school colors, we will have expenses of $100 to rent the Acme PomPom Plant, and then $0.25 per pom pom in materials. We believe that we can sell 500 pom poms if we charge $1.50. Assume the number sold is a linear function of the price. How much should we charge to maximize profit?

I have done this problem many times but I cannot find max profit. I keep fining a max income. I am lost and don't know what to do please help!

  • Calculus Max Profit -

    "Assume the number sold is a linear function of the price"
    but no information is given on the price elasticity, so we will assume, in general, an increase in price of $1 will increase sales by m units.
    Under normal supply-demand curve, m is necessarily negative, of the order of -200 or so.

    With that in mind, and knowing that (500,1.50) is a point on the line sales versus price, we construct the sales(y)-price(x) relation as:
    (y-500)=m(x-1.50)
    therefore, at a price of x, we expect sales of
    y=m(x-1.5)+500

    Total revenue,
    R=xy

    Total profit
    P=xy - cost
    =xy - (0.25x+100)
    =x(m(x-1.5)+500) - (0.25x+100)

    To get the maximum profit, we differentiate profit with respect to price, and equate to zero to find the optimum price:

    dp/dx = m*x+m*(x-1.5)+1999/4 =0
    Solve for x:
    x=(6*m-1999)/(8*m)

    If the price elasticity m=-100,
    x=2599/800=$3.25
    if m=-200
    x=3199/1600=$2
    if m=-300
    x=3799/2400=$1.58
    ...
    etc.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. business

    Please check. Mary Torres owns and operates a bakery. During the past week she sold 500 loaves of bread at $2.25 per loaf. The raw materials for each loaf cost Ms. Torres $1.80. - Aside from the costs from raw materials, what other …
  2. Math

    How do I formulate this question to be solved in the POM-QM software?
  3. Economics

    Assume a monopolist with the following: a. Qd = 100 – 10p b. TC = 1 + 2Q Find the following: Price at profit max Quantity at profit max TR at profit max TC at profit max Profit
  4. POM

    Acme Tire Service completed 100 tire changes, 6 brake jobs, and 16 alignments in an eight hour day with his standard crew of 6 mechanics. A brake specialist cost $16 per hour, a tire changer costs $8 per hour and an alighnment mechanic …
  5. ENGLISH

    ive done ten sentences, five are verb and five are adjective, please check. 1. Lily bought a pink cat mask for the halloween party. 2. Anna was covering her scar with makeup as she didn't want anyone to make fun of it. 3. Tory dressed …
  6. Computer Programming

    I'm trying to create a vba code that solves this problem: 50 apartment units rent for 600 a month, for each 40 dollar increase in rent a unit becomes unoccupied. I want to find the max profit and number of units at that number. Sub …
  7. Algebra 1

    For $46, Joel can rent a machine to make novelty buttons to sell at the county fair. The materials cost $0.39 per button. How many buttons must he sell at $1.40 each in order to make a profit. Identify the graph that shows all the …
  8. English

    1. Read the following paragraph from "Five Hours to Simla." After the cacophony of screeching brakes and grinding gears, there followed the comparatively static hum of engines, and drivers waited in exasperation for the next lurch …
  9. English

    Read the following paragraph from "Five Hours to Simla." After the cacophony of screeching brakes and grinding gears, there followed the comparatively static hum of engines, and drivers waited in exasperation for the next lurch forwards. …
  10. pre-algebra, math

    You make and sell birdhouses. Your fixed costs for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50. Let x represent the number of birdhouses …

More Similar Questions