Determin the cost, in dollars of delivering x parcel per flight

Question

An air freight company has determined that the cost, in dollars, of delivering x parcels per flight is
C(x) = 4025 + 5x.

 

The price per parcel, in dollars, the company charges to send x parcels is
p(x) = 20 − 0.01x.

 

Determine
(a) the revenue function.
R(x) =


(b) the profit function.
P(x) =


(c) the company's maximum profit.
$

(d) the price per parcel that yields the maximum profit.
$

(e) the minimum number of parcels the air freight company must ship to break even.
parcels

Answer 1

  The revenue, or how much money you collect is the number of parcels times the price per parcel.

So, R(x) = x*p(x) = x(20-.01x) = 20x - .01x^2

 

   The profit is income - cost.....so, P(x) = R(x) - C(x) = 20x - .01x^2 - (4025 + 5x). Simplifying you'd get 

P(x) = -.01x^2 +15x - 4025.

 

    If you know calculus, take the derivative, set it = 0, and you'll find the x that leads to the maximum. Since this is labeled college algebra, I will do this with the properties of a parabola. The maximum (or minimum) value of a parabola is always on the axis of symmetry. You can tell if it's a maximum (or minimum) by looking at the coefficient of the x^2 term. Since our value is negative (-.01), we have a "frowning" parabola and so we do have a maximum. (negative frown but positive smile......corny but efficient)

   Axis of symmetry = -b/2a or x= -15/-.02 =750. So 750 parcels leads to the maximum profit.

so maximum profit is P(750)= -.01(750)^2+15(750)- 4025 or 1600 dollars.

   price per parcels is p(750) = 20-.01(750) = 12.50 dollars per parcel.

 

Break even point is when cost = revenue or 4025 +5x = 20x - .01x^2 .

402500 + 500x =2000x -x^2

x^2 - 1500x +402500 = 0

(x-350)(x-1150) = 0

x=350 x=1150

 

So, as long as the number of parcels is between these values you'll make money. If you have fewer than 350 you don't make enough profit to justify the cost. If you have more than 1150, your price per parce is so low that you don't make enough money either.

 

Hope this helps.