Question

A company has learned that when it initiates a new sales campaign, the number of sales per

day increases. However, the number of extra daily sales per day decreases as the impact of the campaign wears off. For a specific campaign the company has determined that if there are e 𝑆(𝑡) extra daily sales as a result of the campaign and 𝑡 days have elapsed since the campaign ended, then 𝑆(𝑡) = 1000 (3^−𝑡/2). Find the rate at which the extra daily sales are decreasing when (a) 𝑡 = 4 and (b) 𝑡 = 10.

Answer 1

$S'(t) = 1000 \times -\frac{1}{2} 3^{-\frac{t}{2}} \ln{3} = -500\ln{3}\left(3^{-\frac{t}{2}}\right)\\ S'(4) = -500\ln{3}\left(3^{-\frac{4}{2}}\right) = -61.03\\ S'(10) = -500\ln{3}\left(3^{-\frac{10}{2}}\right) = -2.26\\ \textsf{Therefore, the rate at which the}\\ \textsf{extra daily sales are decreasing}\\ \textsf{when}\\ (a) t = 4\,\,\textsf{is}\,\, 61.03\,\, \textsf{Sales}/\textsf{day.}\\ (b) t = 10\,\,\textsf{is}\,\, 2.26\,\,\textsf{Sales}/\textsf{day.}\\$