Chapter 7.7, Problem 7P

a.

To determine

To find the total membership money of the Ski club after recruiting 10, 15 and n new members.

Answer to Problem 7P

The total membership money after recruiting 10, 15 and $n$ new members are $\$210$ , $\$227.5$ and $\$\left[\left(20+n\right)\cdot \left(8-0.1n\right)\right]$ respectively.

Explanation of Solution

Given information:

Currently, the number of members in the club is 20. The dues per member are $\$8$ and the dues for all new members be reduced by $10$ cents.

Here, $10\text{ cents}=\frac{10}{100}=\$0.1$ .

After recruiting 10 new members, total number of members in the club $=20+10=30$ .

Total dues reduced after recruiting 10 new members $=10\times 0.1=\$1$ .

After reducing $\$1$ for all members, the new dues per member $=8-1=\$7$ .

Now, the total membership money $=7\times 30=\$210$ .

After recruiting 15 new members, total number of members in the club $=20+15=35$ .

Total dues reduced after recruiting 15 new members $=15\times 0.1=\$1.5$ .

After reducing $\$1.5$ for all members, the new dues per member $=8-1.5=\$6.5$ .

Now, the total membership money $=6.5\times 35=\$227.5$ .

After recruiting $n$ new members, total number of members in the club $=\left(20+n\right)$ .

Total dues reduced after recruiting $n$ new members $=n\times 0.1=\$0.1n$

After reducing $\$0.1n$ for all members, the new dues per member $=\$\left(8-0.1n\right)$ .

Now, the total membership money $=\$\left[\left(20+n\right)\cdot \left(8-0.1n\right)\right]$ .

Hence,

The total membership money after recruiting 10, 15 and $n$ new members are $\$210$ , $\$227.5$ and $\$\left[\left(20+n\right)\cdot \left(8-0.1n\right)\right]$ respectively.

b.

To determine

To find the value of $n$ for which the total membership money of the club will be maximum.

Answer to Problem 7P

The total membership money of the club will be maximum, when the number of new members is 30.

Explanation of Solution

Given information:

Currently, the number of members in the club is 20. The dues per member are $\$8$ and the dues for all new members be reduced by $10$ cents.

The standard form of a quadratic equation: $y=a{x}^2+bx+c$ .

So, the formula of the axis of symmetry: $x=\frac{-b}{2a}$ .

Let, the total membership money after recruiting $n$ new members is $\$y$ .

  $\therefore y=(20+n)\cdot (8-0.1n)=160-2n+8n-0.1n^2=-0.1n^2+6n+160$

Now, comparing it with the standard form of the quadratic equation:

  $a=-0.1,b=6\text{ and }c=160$ .

By applying the formula of the axis of symmetry:

  $n=\frac{-b}{2a}=\frac{-6}{2\cdot (-0.1)}=\frac{3}{0.1}=30$ .

Here, $n$ represents the number of new members for maximum membership money.

Hence,

The total membership money of the club will be maximum, when the number of new members is 30.