Answer to Question #153679 in Calculus for Wendel

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) Rate of cange of S is a derivative of S

$$S'=(4000 lnx)'=4000/x$$

If x = 800, then S' = 4000/800 = 5

So, the rate of change of sales income with respect to advertising expense when 800 is the weekly advertising budget is $5.

(b) Let x₀ = 800, x₁ = 950. Find the S₁ - S₀.

$$S_1≈S_0+S'(x_0)*(x_1-x_0)$$ $$S_1-S_0≈S'(x_0)*(x_1-x_0)$$ $$S_1-S_0≈5*(950-800)=5*150=750$$

So, the approximate increase in the total weekly income from sales is $750.