Answer:
The value is [tex]y = \$ 225[/tex]
Step-by-step explanation:
From the question we are told that
The amount charge per year is [tex]k = \$ 200[/tex]
The number of members it will have at this amount is [tex]n = 50[/tex]
The amount amount increase that will lead to the loss of a single member is [tex]z = \$ 5[/tex]
Generally the total amount the club would obtain from its members is mathematically represented as
[tex]I = Amount \ due\ paid * Number \ of members[/tex]
Now let x denote the number of member lost
Hence
[tex]I = (k + zx) (n-x )[/tex]
=> [tex]I = (200 + 5x) (50-x )[/tex]
=> [tex]I = 10000+50x-5x^2[/tex]
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
[tex]\frac{dI}{dx} = 50-10x[/tex]
=> [tex] x = 5[/tex]
So from [tex]I = (200 + 5x) (50-x )[/tex] we have
[tex]I = (200 + 5 (5)) (50-5 )[/tex]
[tex]I = \$ 10125 [/tex]
So the amount the club should charge is
[tex]y = \frac{10125}{50 - 5}[/tex]
[tex]y = \$ 225[/tex]
