The sales $S$ (in thousands of units) of a cleaning solution after $x$ hundred dollars is spent on advertising are given by $S = 20 \left( 1 - e ^ { k x } \right)$. When $\$ 450$ is spent on advertising, 2500 units are sold. Complete the model by solving for $k$ and use the model to estimate the number of units that will be sold if advertising expenditures are raised to $\$ 650$. Round your answer to the nearest unit.
A) 3508 units
B) 6401 units
C) 5508 units
D) 2483 units
E) 8787 units

Question

Quizplus · Accepted Answer

When $x = 4.5$, $S = 2500$. So,$$2500 = 20 \left( 1 - e ^ { 4.5 k } \right)$$$$\Rightarrow \qquad 1 - e ^ { 4.5 k } = \frac{2500}{20} = 125$$$$\Rightarrow \qquad e ^ { 4.5 k } = -124$$$$\Rightarrow \qquad 4.5 k = \ln (-124)$$$$\Rightarrow \qquad k = \frac{\ln (-124)}{4.5} \approx -0.277$$So, the model is $S = 20 \left( 1 - e ^ { -0.277 x } \right)$.When $x = 6.5$,$$S = 20 \left( 1 - e ^ { -0.277 (6.5) } \right) \approx 3508$$Therefore, the correct answer is A.

The Sales SSS (In Thousands of Units) of a Cleaning Solution After

The Sales $S$ (In Thousands of Units) of a Cleaning Solution After