Question

A particular company has determined that when its weekly advertising expense is 𝑥 dollars,

then if 𝑆 dollars is its total weekly income from sales, 𝑆 = 4000 𝑙𝑛𝑥.

(a) Determine the rate of change of sales income with respect to advertising expense when $800 is the weekly advertising budget.

(b) If the weekly advertising budget is increased to $950 dollars, what is the approximate increase in the total weekly income from sales?

Answer 1

$a)S(x) = 4000\ln{x}$

$\text {rate of change of sales }S'(x)$

$S'(x)=(4000\ln(x))'= \frac{4000}{x}$

$S'(800)=\frac{4000}{800}=5$

$\text{(b) total weekly income from sales for advertising expense =800}$

$S_1= 4000\ln(800)\approx26738$

$\text{for advertising expense =950}$

$S_1= 4000\ln(950)\approx27425$

$S_2-S_1=27425-26738 =687$

Answer:(a) 5 ; (b) 687