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Calculus, EXT1 C3 EQ-Bank 7

The rate of weekly sales of a product that has just been released can be modelled by the following differential equation

`S^(′)(t)=500/(t+1)^3-250/(t+1)^2`

where `S` is the number of sales in thousands and `t` is the number of weeks since launch.

Good quality products will result in increasing sales.

  1. Find the function that describes the weekly sales.   (2 marks)

  2. By calculating the number of sales for the first and fifth week, comment on the results of the launch with respect to the given information.  (2 marks)

Show Answers Only
  1. `S(t)=250/(t+1)-250/((t+1)^2)`
  2. `S(1) = 62\ 500, \ S(5) = 34\ 722`
  3. `text{Sales drop significantly between week 1 and week 5, suggesting}`
  4. `text{the product is of poor quality.}`
Show Worked Solution

i.   `S^(′)(t)=500/(t+1)^3-250/(t+1)^2`

`S(t)` `=int 500/(t+1)^3-250/(t+1)^2\ dt`  
  `=int 500(t+1)^(-3)-250(t+1)^(-2)\ dt`  
  `=(500(t+1)^(-2))/(-2)-(250(t+1)^(-1))/(-1)+c`  
  `=250/(t+1)-250/((t+1)^2)+c`  

 
`text{When}\ \ t=0, S(t)=0:`

`0=250/1-250/(1^2)+c\ \ =>\ \ c=0`

`:.S(t)=250/(t+1)-250/((t+1)^2)`
 

 ii.   `S(1)=250/(1+1)-250/((1+1)^2)=62.5`

`=>\ text{Sales in week 1 = 62 500}`
 

`S(5)=250/(1+5)-250/((1+5)^2)=34.7222`

`=>\ text{Sales in week 5 = 34 722}`
 

`text{Sales drop significantly between week 1 and week 5, suggesting}`

`text{the product is of poor quality.}`