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

The value is
`y  =  \$ 225`

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

`(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  =  (10125)/(50 - 5)`

`y  =  \$ 225`