Question

1. get fit®, power house®, and fitness frenzy® are competing gyms that are trying to recruit new members. they all have new deals to attract customers.

• get fit: $35 one - time registration fee and $5 for every time you visit
• power house: $40 one - time registration fee and $4 for every time you visit
• fitness frenzy: no sign - up fee! pay $15 a month.

consider the graphs that represent the cost of attending a gym for the first month if you visit x times, for each gym.
a. which gyms graph, if any, would represent a horizontal line? explain.
b. which gyms graphs, if any, would represent perpendicular lines? explain.
c. come up with a pricing scheme for a new gym, run around the clock®, so that the graph of monthly cost of attending the gym x times is parallel to that of power house®.

Explanation:

Step1: Write cost - function for each gym

Let $x$ be the number of visits in a month.
For Get Fit, the cost function $C_{1}(x)=35 + 5x$.
For Power House, the cost function $C_{2}(x)=40+4x$.
For Fitness Frenzy, the cost function $C_{3}(x)=15$ (since it's a flat - rate per month and not dependent on the number of visits).

Step2: Analyze horizontal line

A horizontal line has a slope of 0. The cost function of Fitness Frenzy, $C_{3}(x)=15$, has a slope of 0. So, its graph is a horizontal line because the cost does not depend on the number of visits $x$.

Step3: Analyze perpendicular lines

The slope of the line for Get Fit is $m_1 = 5$, the slope of the line for Power House is $m_2=4$, and the slope of the line for Fitness Frenzy is $m_3 = 0$. Two lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1\times m_2=- 1$. Since none of the products of the slopes of the given lines equal - 1, none of the gyms' graphs represent perpendicular lines.

Step4: Find parallel - pricing scheme

Two lines are parallel if they have the same slope. The slope of the line for Power House is 4. A pricing scheme for Run Around the Clock could be a one - time registration fee of $20 and $4 for every time you visit. The cost function would be $C(x)=20 + 4x$, which has the same slope as the cost function of Power House.

Answer:

a. Fitness Frenzy. The cost function $C(x)=15$ is a constant function, so its graph is a horizontal line as the cost does not depend on the number of visits $x$.
b. None of them. The slopes of the lines representing the cost functions of the given gyms do not satisfy the condition $m_1\times m_2=-1$ for perpendicular lines.
c. A one - time registration fee of $20 and $4 for every visit. The cost function is $C(x)=20 + 4x$.