At what rate (in dollars per thousand dollars of advertising) are Cannon's total sales increasing when the amount of money spent on advertising

Question

The relationship between the amount of money x that Cannon Precision Instruments spends on advertising and the company's total sales S(x) is given by the following function where S(x) and x are both measured in thousands of dollars.

S(x) = −0.002x³ + 0.9x² + 8x + 500      (0 ≤ x ≤ 200)

Find S '(x), the rate of change of the sales with respect to the amount of money spent on advertising.

S '(x) =  

−0.006x2+1.8x+8

 

 

At what rate (in dollars per thousand dollars of advertising) are Cannon's total sales increasing when the amount of money spent on advertising is $120,000?

$  per thousand dollars of advertising

At what rate (in dollars per thousand dollars of advertising) are Cannon's total sales increasing when the amount of money spent on advertising is $170,000?

$ per thousand dollars of advertising

Answer 1

So this is a little tricky to think about, because you normally would think of advertising as some unknown future sales for a company, but we have some equations here that model advertising to sales nicely for Cannon Precision Instruments.

You did the derivative correctly of the equation above:

S(x) = −0.002x³ + 0.9x² + 8x + 500  (0 ≤ x ≤ 200), with the derivative as:

S '(x) =  −0.006x2+1.8x+8

The derivative here signifies the rate that the sales for Cannon is changing. The (0 ≤ x ≤ 200) above is the range of advertising dollars spent (in thousands) that the equations is valid for.

To answer the question of "What rate are cannon's sales increasing at $120k and $170k" we can use the derivative equation as S'(120) and S'(170), so:

S'(120) = −0.006(120)²+1.8(120)+8 = (-86.4) + 216 + 8 = 137.6

S'(170) = −0.006(170)²+1.8(170)+8 = (-173.4) + 306 + 8 = 140.6

Don't forget to convert these back to thousands, so:

When cannon is spending 120,000 on advertising (S'(120)), the rate of sales are increasing at $137,600 dollars, and

When cannon is spending 170,000 on advertising (S'(170)), the rate of sales are increasing at $140,600.