Question

If a club charges dues of $200 a year, it will have 50 members. For each $5 it raises its dues, it loses a member. USE AN EQUATION to determine what the club should charge to maximize its income from dues.

Answer 1

Answer:

The value is      $y = \ 225$

Step-by-step explanation:

From the question we are told that

   The amount charge per year is  $k = \ 200$

   The  number of members it will have at this amount is  $n = 50$

   The amount amount increase that will lead to the loss of a single member is   $z = \ 5$

        Generally the total amount the club would obtain from  its members is mathematically represented as

      $I = Amount \ due\ paid * Number \ of members$

Now let x denote the number of member lost

Hence

      $I = (k + zx) (n-x )$

=>    $I = (200 + 5x) (50-x )$

=>  $I = 10000+50x-5x^2$

Thus the number of members that be removed to  give the maximum  income from dues is obtained by differentiating the above equation and equating it to zero

           $\frac{dI}{dx} = 50-10x$

=>         $x = 5$

So from  $I = (200 + 5x) (50-x )$ we have

          $I = (200 + 5 (5)) (50-5 )$

           $I = \ 10125$

So the amount the club should charge is    

         $y = \frac{10125}{50 - 5}$

         $y = \ 225$