Daily sales approximately follow the formula s = 2d^2 - 740d + 69000

Question

where d is the day of the year (a number between 1 and 366) and s is the sales volume (in thousands of dollars). According to this formula, what’s the minimum daily sales, and on what day of the year will it occur? First type the minimum daily sales (a number, representing thousands of dollars), then type a /, then type the day it occurs (as a number). Do not type any spaces.

Answer 1

s = 2d^2 - 740d + 69000

 

This is a max/min problem. To find an extremum, one wants to take the first derivative and set it equal to 0. It is unfortunate that the dependent variable is given by d. But, the first derivative is

 

ds/dd = 4d - 740 = 0

 

So, 4d = 740

or

 

d = 185

 

Is this a maximum or a minimum?

 

Take a second derivative

d^2 s/ dd^2 = 4 > 0, so it is concave up or a minimum.

 

On day 185, s(185) = 2*(185)^2 - 740(185) + 69000 = 550

 

Answer: $550,000/185