Question

An advertising company designs a campaign to introduce a new product to a metropolitan area of population 1 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 20 days of advertising. The number of people who become aware of the product at time t is: a) P(t) = 1 - e^{-frac{5}{2}t} b) P(t) = 1 - (frac{1}{2})^{frac{1}{20}t} c) P(t) = e^{-frac{1}{20}t} d) P(t) = (frac{1}{2})^{-frac{1}{20}t} e) P(t) = 1 - e^{-frac{1}{20}t} f) None of the above.

          An advertising company designs a campaign to introduce a new product to a metropolitan area of population 1 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 20 days of advertising. The number of people who become aware of the product at time t is:
a) P(t) = 1 - e^{-frac{5}{2}t}
b) P(t) = 1 - (frac{1}{2})^{frac{1}{20}t}
c) P(t) = e^{-frac{1}{20}t}
d) P(t) = (frac{1}{2})^{-frac{1}{20}t}
e) P(t) = 1 - e^{-frac{1}{20}t}
f) None of the above.

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$An advertising company designs a campaign to introduce a new product to a metropolitan area of population 1 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 20 days of advertising. The number of people who become aware of the product at time t is: a) P(t) = 1 - e^-frac52t b) P(t) = 1 - (frac12)^frac120t c) P(t) = e^-frac120t d) P(t) = (frac12)^-frac120t e) P(t) = 1 - e^-frac120t f) None of the above.$

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Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Jacob Fry

09/15/2022

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An advertising company designs a campaign to introduce a new product to a metropolitan area of population 1 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 20 days of advertising. The number of people who become aware of the product at time t is: a) P(t) = 1 - e^(-5/2 t) b) P(t) = 1 - (1/2)^(1/20 t) c) P(t) = e^(-1/20 t) d) P(t) = (1/2)^(-1/20 t) e) P(t) = 1 - e^(-1/20 t) f) None of the above.

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Transcript

00:01 So in this question, we're told that we have a total population of the 1 million people.

00:07 So p total equals 1 in units of millions.

00:15 Now, p of t is the number of people who are aware of the product, who become aware of the product by time t.

00:28 So the total number of people who are aware by time t.

00:42 Now we're told that p increases a rate proportional to the number of people who are still unaware of the product.

00:55 So the number of people still unaware of the product is 1 minus p of t, and this is just a proportionality constant.

01:03 Now we're also told that nobody was aware of the product at the beginning of the campaign, and that after 20 days of advertising, 50 % of people were aware.

01:18 So now we can solve this differential equation.

01:21 So the integral of p over 1 minus p is going to be equal to the integral of adt.

01:33 And at time t equals zero, nobody is aware of the product.

01:39 And then at time t equals t, we're going to have p of t.

01:44 People are going to be aware.

01:47 So let's integrate this.

01:49 So this is going to be log of 1 minus p.

01:55 With a minus sign because the derivative of this is going to be minus 1 over 1 minus p.

02:01 Now putting in these two boundary conditions, it's just the same as evaluating this at each of the boundary conditions and dividing them.

02:13 But i'll save that for a minute...

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