Question

A store finds that its sales revenue changes at a rate given by S'(t) = −30t2 + 420t dollars per day where t is the number of days after an advertising campaign ends and 0 ≤ t ≤ 30. (a) Find the total sales for the first week after the campaign ends (t = 0 to t = 7). $ (b) Find the total sales for the second week after the campaign ends (t = 7 to t = 14). $ g

Answer 1

Explanation:

Give the rate of change of sales revenue of a store modeled by the equation
`S'(t)= -30t^(2) + 420t`. The Total sales revenue function S(t) can be gotten by integrating the function given as shown;

`\int\limits {S'(t)} \, dt = \int\limits ({-30t^(2)+420t }) \, dt \\S(t) = (-30t^(3) )/(3)+(420t^(2) )/(2)\\ S(t)= -10t^(3) +210t^(2) \\`

a) The total sales for the first week after the campaign ends (t = 0 to t = 7) is expressed as shown;

`Given\ S(t) = -10t^(3) + 210t^(2)`

`S(0) = -10(0)^(3) + 210(0)^(2)\\S(0) = 0\\S(7) = -10(7)^(3) + 210(7)^(2)\\S(7) = -3430+10,290\\S(7) = 6,860`

Total sales = S(7) - S(0)

= 6,860 - 0

Total sales for the first week = $6,860

b) The total sales for the secondweek after the campaign ends (t = 7 to t = 14) is expressed as shown;

Total sales for the second week = S(14)-S(7)

Given S(7) = 6,860

To get S(14);

`S(14) = -10(14)^(3) + 210(14)^(2)\\S(14) = -27,440+41,160\\S(14) = 13,720`

The total sales for the second week after campaign ends = 13,720 - 6,860

= $6,860