A model for product awareness as a result of an advertising campaign is given by

Question

A model for product awareness as a result of an advertising campaign is given by A()49.3+ 23.5 In(2 20) where z is the number of days since the start of the campaign and A(x) is the percentage of the demographic that is aware of the product. 1. What percentage of the target population is aware of this product 10 days after the start of the advertising campaign? Use correct units in your answer 2. What percentage of the target population is aware of this product 40 days after the start of the advertising campaign? Use correct units in your answer 3. Find the formula for the rate of change of the population that is aware of this product.

Explanation / Answer

A(x) = - 49.3 + 23.5 ln (2x+20)
1) So, after 10 days of start of advertising campaign, for x=10 we get
A(x) = A(10) = - 49.3 + 23.5 ln (2*10 + 20) = - 49.3 + 23.5 ln 40 = - 49.3 + 23.5 * 3.688
= - 49.3 + 86.67 = 37.37
Thus, 10 days after the start of the ad campaign, 37.7% of demographic is aware of the product;

2) for 40 days after the start of the ad campaign;
A(x) = A(40) = -49.3 + 23.5 ln (2*40+20) = - 49.3 + 23.5 ln 100 = -49.3 + 23.5 (4.605) = -49.3 + 108.22=58.92
Thus, 40 days after the start of advertising, 58.92% of demographic is aware of the product;

3) Rate of change of population that is aware of this product (can be calcualted in terms of % of population) as A(x) is also in % terms;
A(x) = - 49.3 + 23.5 ln (2x+20)
Rate of change of population aware of the product ; dA(x) / dx
= 23.5 / ln(2x+20) * 2 = 47 / ln(2x+20)
Thus, rate of change of population that is aware of the product = 47/ ln(2x+20)

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