Chapter 7.7, Problem 8P

To determine

To find the number of new members and the total membership money of the club.

Answer to Problem 8P

The total membership money after recruiting 28 new members is $\$270.4$ .

Explanation of Solution

Given information:

Currently, the number of members in the Ski club is 24. The dues per member are $\$8$ and the dues for all new members be reduced by $10$ cents.

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

Let, the number of new members $=x$ and the total membership money $=y$ .

After recruiting $x$ new members, total number of members in the club $=\left(24+x\right)$ .

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

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

Now, the total membership money $=\$\left[\left(24+x\right)\cdot \left(8-0.1x\right)\right]$ .

  $\therefore y=\left(24+x\right)\cdot \left(8-0.1x\right)=192+5.6x-0.1x^2$ .

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

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

Now, comparing the equation of the total membership money with the standard form of the quadratic equation:

  $a=-0.1,b=5.6\text{ and }c=192$ .

By applying the formula of the axis of symmetry:

  $x=\frac{-b}{2a}=\frac{-5.6}{2\cdot (-0.1)}=28$ .

  $\therefore y=192+5.6x-0.1x^2=192+5.6\times 28-0.1\times {\left(28\right)}^2=192+156.8-78.4=270.4$ .

Hence,

The total membership money after recruiting 28 new members is $\$270.4$ .