Algebra and Trigonometry: Structure and Method, Book 2

2000th Edition

ISBN: 9780395977255

Author: MCDOUGAL LITTEL

Publisher: McDougal Littell

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Question

Chapter 7.7, Problem 7P

To determine

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

Expert Solution

Answer to Problem 7P

The total membership money after recruiting 10, 15 and n new members are $210 , $227.5 and $[(20+n)⋅(8−0.1n)] 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 cents=10100=$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 =10×0.1=$1 .

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

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

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

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

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

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

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

Total dues reduced after recruiting n new members =n×0.1=$0.1n

After reducing $0.1n for all members, the new dues per member =$(8−0.1n) .

Now, the total membership money =$[(20+n)⋅(8−0.1n)] .

Hence,

The total membership money after recruiting 10, 15 and n new members are $210 , $227.5 and $[(20+n)⋅(8−0.1n)] respectively.

To determine

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

Expert Solution

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=ax2+bx+c .

So, the formula of the axis of symmetry: x=−b2a .

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

∴y=(20+n)⋅(8−0.1n)=160−2n+8n−0.1n2=−0.1n2+6n+160

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

a=−0.1, b=6 and c=160 .

By applying the formula of the axis of symmetry:

n=−b2a=−62⋅(−0.1)=30.1=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.

Chapter 7.7, Problem 6P

Chapter 7.7, Problem 8P

Chapter 7 Solutions

Algebra and Trigonometry: Structure and Method, Book 2

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