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. Number NPrice p25052.5030052.0035051.5040051.00 (a) Find a formula for p in terms of N modeling the data in the table. p= (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=

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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.  Number NPrice p25052.5030052.0035051.5040051.00  (a) Find a formula for p in terms of N modeling the data in the table.  p=  (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=

Nathaniel Kramer · Accepted Answer

Slope of line joining the points (250,52.50)and(300,52.00)is m=y2−y1x2−x1=52.00−52.50300−250=−0.01 Slope of line joining the points (300,52.00)and(350,1.505)is m=y2−y1x2−x1=51.50−52.00350−300=−0.01 and so on. Since slope of line joining the points (250, 52.50) and (300, 52.00) is equal to the slope of line joining the points (300, 52.00) and (350, 1.505) , so there is a linear relation between p and N Let p=mN+b where m=−0.01 Therefore, p=−0.01N+b Now, the point (250, 52.50) lie on the line, so 52.50=−0.01×250+b 52.50=−2.5+b b=55 Thus, p=−0.01N+55 (b) 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 is determined as follows R=pN R=(−0.01N+55)N R=−0.01N2+55N Finally answer: (a)p=−0.01N+55 (b) R=−0.01N2+55N

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

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