Question

The publisher of a recently released nonfiction book expects that over the first 20 months after its release, the monthly profit can be approximated with the model P(t) 240t - 40t t2 +20 where P is in thousands of dollars and t is in months from the time of release. a. Derive a model that gives the marginal profit of the book. P'(t) Preview b. Specify the expected marginal profit 5 months after release. Round to 2 decimal places thousands of dollars per month c. Specify the expected marginal profit 11 months after release. Round to 2 decimal places. thousands of dollars per month d. After which month is the monthly profit start to decrease steadily?

Answer 1

(a)[TeX]\frac{dP}{dt}=\frac{4800-1600t-240t^2}{(t^2+20)^2}[/TeX]

(b)P'(5)=-($4.54) Thousand

(c)P'(11)=-($2.10) Thousand

(d)The fifth Month

Explanation:

Given the monthly profit model:

[TeX]P(t)=\frac{240t-40t^2}{t^2+20}[/TeX]

(a)We want to derive a model that gives the Marginal Profit, P' of the book.

We differentiate

[TeX]P(t)=\frac{240t-40t^2}{t^2+20}[/TeX] using quotient rule.

[TeX]\frac{dP}{dt}=\frac{(t^2+20)(240-80t)-(240t-40t^2)(2t)}{(t^2+20)^2}[/TeX]

Simplifying

[TeX]\frac{dP}{dt}=\frac{4800-1600t-240t^2}{(t^2+20)^2}[/TeX]

We have derived a model for the marginal profit.

(b) After 5 months, at t=5

Marginal Profit=P'(5)

[TeX]\frac{dP}{dt}=\frac{4800-1600t-240t^2}{(t^2+20)^2}[/TeX]

[TeX]P^{'}(5)=\frac{4800-1600(5)-240(5)^2}{(5^2+20)^2}[/TeX]

=-($4.54) Thousand of dollars

(c)Marginal Profit 11 Months after book release

[TeX]P^{'}(11)=\frac{4800-1600(11)-240(11)^2}{(11^2+20)^2}[/TeX]

=-($2.10) Thousand of dollars

(d) Since the marginal profit at t=5 is negative, after the 5th Month, the profit starts to experience a steady decrease.