Question

There are currently 20 members in a school Ski club and the dues are $8 per member. To encourage the recruiting of new members, the club treasurer suggests that for each new member recruited, the dues foe all members be reduced by 10 cents. How many new members will result in the maximum membership money? How much will that be?

Answer 1

The maximum membership money for the Ski club is obtained when they recruit 2 new members, yielding $171.60. They achieve this by setting up a dues structure that decreases the dues by $0.10 for each new member recruited.

Step-by-step explanation:

The school Ski club is trying to maximize the membership money with a plan that reduces the dues for every member by $0.10 for each new member recruited. Let's denote x as the number of new members. Initially, there are 20 members, so with the new members, there will be 20 + x members. The original dues are $8 per member, but they are reduced by $0.10 per new member; thus, the new dues will be $8 - $0.10x per member.

To find the maximum membership money, we need to express the total money, M, collected as a function of x:

M(x) = (20 + x) × ($8 - $0.10x)

Expanding the equation gives us:

M(x) = 160 + 8x - 2x²

This is a quadratic equation, and its graph is a parabola opening downwards, indicating that the maximum value of M occurs at the vertex of the parabola. The x-coordinate of the vertex is given by -b/(2a), where a is the coefficient of x² (which is -2 in this case) and b is the coefficient of x (which is 8).

The number of new members that will result in the maximum membership money is:

x = -8 / (2(-2)) = 2

Hence, the Ski club should recruit 2 new members to maximize the total money. Calculating the total dues at this point:

M(2) = (20 + 2) × ($8 - $0.10 × 2) = 22 × $7.80 = $171.60

This is the maximum amount of membership money the club can earn by restructuring the dues.

Answer 2

How many new members will result in the maximum membership money?

n=30

Step-by-step explanation:

the expression for membership money is ___ (members) * (dues) = money

where n is the number of new members ___ (20 + n)(8 - .1n) = m ___ 160 + 6n - .1n^2 = m

a) substitute the values for new members and solve for m

b) the maximum lies on the axis of symmetry of the quadratic

___ the general formula of the axis of symmetry for ax^2+bx+c is ___ x = -b / (2a)

in this case ___ n = -6 / [2(-.1)] ___ this is the number of new members for maximum money

n=30