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Writing Quadratic Functions and Equations: Mastery Test

Select the correct answer.

A fitness center currently has 320 members. Monthly membership fees are [tex]$\$[/tex]45[tex]$. The manager of the fitness center has determined that each time membership fees increase by $[/tex]\[tex]$5$[/tex], approximately 10 members leave and go to a different gym.

Write an equation that can be used to find the revenue of the fitness center in dollars, [tex]$y$[/tex], after [tex]$x$[/tex] price increases of [tex]$\$[/tex]5[tex]$.

A. $[/tex]y = -50x^2 + 3,425x + 14,400[tex]$
B. $[/tex]y = -50x^2 + 1,150x + 14,400[tex]$
C. $[/tex]y = -50x^2 + 2,975x + 14,400[tex]$
D. $[/tex]y = -50x^2 + 2,050x + 14,400$



Answer :

GinnyAnswer
To determine the equation that represents the revenue of the fitness center after [tex]\( x \)[/tex] price increases of [tex]\( \$5 \)[/tex], let's break it down step by step.

1. Understand the initial conditions:
- Initial number of members: [tex]\( 320 \)[/tex]
- Initial membership fee per member: [tex]\( \$45 \)[/tex]

2. Determine the effects of each price increase:
- Each [tex]\( \$5 \)[/tex] increase causes 10 members to leave the fitness center.

3. Generalize the variables:
- Let [tex]\( x \)[/tex] be the number of [tex]\( \$5 \)[/tex] increases.

4. Updated membership fee:
- If the original fee was [tex]\( \$45 \)[/tex], and each increase is [tex]\( \$5 \)[/tex], after [tex]\( x \)[/tex] increases, the new fee will be:
[tex]\[ \text{New fee} = 45 + 5x \][/tex]

5. Updated number of members:
- If 10 members leave for each [tex]\( \$5 \)[/tex] increase, after [tex]\( x \)[/tex] increases, the remaining number of members will be:
[tex]\[ \text{New number of members} = 320 - 10x \][/tex]

6. Revenue equation:
- Revenue is calculated by multiplying the number of members by the membership fee. Therefore, the revenue [tex]\( y \)[/tex] after [tex]\( x \)[/tex] increases is:
[tex]\[ y = (\text{New number of members}) \times (\text{New fee}) \][/tex]
Substituting the values we determined:
[tex]\[ y = (320 - 10x) \times (45 + 5x) \][/tex]

7. Expand the equation:
- To find our correct option, we need to expand and simplify this equation:
[tex]\[ y = (320 - 10x)(45 + 5x) \][/tex]
[tex]\[ y = 320 \cdot 45 + 320 \cdot 5x - 10x \cdot 45 - 10x \cdot 5x \][/tex]
[tex]\[ y = 14400 + 1600x - 450x - 50x^2 \][/tex]
[tex]\[ y = 14400 + 1150x - 50x^2 \][/tex]
Rearranging the terms, we get:
[tex]\[ y = -50x^2 + 1150x + 14400 \][/tex]

Therefore, the correct answer is:

B. [tex]\( y = -50x^2 + 1150x + 14400 \)[/tex]

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