In general, the highest price p per unit of an item at which a manufacturer can sell N items is not constant but is, rather, a function of N. Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
(a) Find a formula for p in terms of N modeling the data in the table.
p=
Number N Price p
250 32.50
300 32.00
350 31.50
400 31.00
(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
R =
(c) On the basis of the tables in this exercise and using cost, C = 30N + 700, use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month.
p=
+6251 I'm going to assume it's linear and then check.(p−32.50)=m(N−250)m=31−32.5400−250=−1.5150=−1100so (p−32.50)=−1100(N−250)p=−N100+35
Checking the two inside points.32=−3+35⇒True31.50=−3.5+35⇒Trueso all the points check out and the formula is correct.
(b) R(N)=Np(N)=N(35−N100)
(c) Profit(N)=R(N)−C(N)=N(35−N100)−(30N+700)
Profit(N)=5N−N2100−700
+6251 I'm going to assume it's linear and then check.(p−32.50)=m(N−250)m=31−32.5400−250=−1.5150=−1100so (p−32.50)=−1100(N−250)p=−N100+35
Checking the two inside points.32=−3+35⇒True31.50=−3.5+35⇒Trueso all the points check out and the formula is correct.
(b) R(N)=Np(N)=N(35−N100)
(c) Profit(N)=R(N)−C(N)=N(35−N100)−(30N+700)
Profit(N)=5N−N2100−700
