A retailer has been selling 1200 tablet computers a week at $350 each.?

2 Answers

• For every $10 discount, 80 more are sold.

  So if the original price of $350 sold 1200 per week, then if we set "x" to be the number of $10 discounts, then the new price becomes:

  (350 - 10x)

  And the number of units sold will be:

  (1200 + 80x)

  the revenue is the product of the cost per unit and the number of units sold, so we can solve for that in terms of x:

  R(x) = (350 - 10x)(1200 + 80x)

  R(x) = 420000 + 28000x - 12000x - 800x²

  R(x) = 420000 + 16000x - 800x²

  To find the maximum revenue, we can solve for the zero of the first derivative to get the number of $10 discounts to apply:

  R'(x) = 16000 - 1600x

  0 = 16000 - 1600x

  1600x = 16000

  x = 10

  So if ten $10 discounts are to be applied to get the maximum revenue, then the cost becomes:

  350 - 10 * 10

  350 - 100

  $250 per unit

  Let's determine how many units will be sold at this price:

  1200 + 80x

  1200 + 80 * 10

  1200 + 800

  2000 units

  -----

  We are then given the cost function:

  C(x) = 35000 + 120x

  Presuming that is in terms of units produced, so before we can find the profit function, we need to re-work our revenue function to also be in terms of units sold (our first one was in terms of $10 discounts).

  Let's start with the vertex form of a quadratic:

  R(x) = a(x - h)² + k

  Where the vertex is the point (h, k). We know this as we solved for the price and units sold in part 1. At a price of $250 each, you sell 2000 units, which is a total revenue of $500,000

  So the vertex is the point (2000, 500000). That is our h and k.

  We also have another point given the original baseline of 1200 units sold at $350, which has a revenue of $420,000), so another data point on the curve is (1200, 420000) That is our x and y (or R(x)). We can substitute what we know and solve for a, then expand it to determine our quadratic:

  420000 = a(1200 - 2000)² + 500000

  -80000 = a(-800)²

  -80000 = 640000a

  a = -80000 / 640000

  a = -1/8

  So our equation is:

  R(x) = (-1/8)(x - 2000)² + 500000

  Expand that out to get:

  R(x) = (-1/8)(x² - 4000x + 4000000) + 500000

  R(x) = (-1/8)x² + 500x - 500000 + 500000

  R(x) = (-1/8)x² + 500x

  Now we have a revenue function in terms of units sold and a cost function in terms of units produced. Presuming you sell all that you manufacture, the profit is the difference between the two :

  P(x) = R(x) - C(x)

  Substitute what we know to come up with the profit function:

  P(x) =(-1/8)x² + 500x - (35000 + 120x)

  and simplify:

  P(x) =(-1/8)x² + 500x - 35000 - 120x

  P(x) =(-1/8)x² + 380x - 35000

  So what is the maximum profit? Do the same as above and solve for the zero of the first derivative:

  P'(x) = (-1/4)x + 380

  0 = (-1/4)x + 380

  (1/4)x = 380

  x = 380 * 4

  x = 1520

  To get the maximum profit, you would want to manufacture and sell 1,520 units.

  We want the price for this level of sales, so let's go back to our original equations and find the number of $10 discounts that are needed to make this level of sales, then work back to find the cost at that level:

  1200 + 80x = 1520

  80x = 320

  x = 4

  So four $10 discounts are needed, making the price:

  350 - 4 * 10

  350 - 40

  $310 per unit.

• a) Find the demand function.

  x-1200 = the increase of units a week per $10 decrease

  80 more units are sold per $10 decrease, therefore,

  p(x) = 350-(10/80)(x-1200)

  = 200-1/8x <-simplified

  b) What should the price be set at in order to maximize revenue?

  R(x) = xp(x)

  =x(200-1/8x)

  =200x-(1/8)x^2

  R'(x)= 200-(1/4)x

  x=800 <- R'(x)=0; solve for x

  Maximized revenue at 800 more units sold

  x-1200=800

  p(x)=350-(10/80)(800) <-substitute 800 for (x-1200)

  = 350-100

  =$250

  c) If the retailer's weekly cost function is

  C(x) = 35,000 + 120x

  what price should it choose in order to maximize its profit?

  P(x) = profit

  P(x)= R(x)-C(x)

  = 200x-(1/8)x^2-(35000+120x)

  = 200x-(1/8)x^2-35000-120x

  = (1/8)x^2-80x+35000 <- simplified and made x^2 term positive

  P'(x)=(1/4)x-80

  x=240 <- P'(x)=0; solve for x

  Maximized profit as 240 units sold

  x-1200=240

  p(x)=350-(10/80)(240) <-substitute 240 for x-1200

  = 350-40

  =$310