mr. saffer is signing up for the gym. he checks out planet fitness and la fitness to compare their costs. planet fitness charges a $50 sign - up fee and then $25 per month. la fitness charges a $20 sign - up fee and $29 per month.
a. write a system of equations that models this situation, where y represents the total cost for x months of having a membership. 2 points
b. if mr. saffer wants to have this gym membership for 1 year at the lowest cost, which gym should he choose? justify your response. 2 points
3. mcdonalds is advertising on snapchat and on instagram. ads on snapchat cost $400 each. ads on instagram cost $60 each. they can run a maximum of 16 total ads. your company has told you to spend at least $2,400 on ads.
a. write a system of inequalities that represents x, the number of snapchat ads and y, the number of instagram ads, they could run. 2 points
b. graph the system of inequalities on the axes provided. label the solution set with an s. 2 points
c. state one combination of ads on snapchat and on instagram that would allow you to meet all requirements. justify your response. 2 points

Question

Accepted Answer

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### First Problem (Gym Memberships)
# Explanation:
## Step1: Define Planet Fitness cost
Total cost = sign-up fee + monthly cost × months.
$y = 50 + 29x$
## Step2: Define LA Fitness cost
Total cost = sign-up fee + monthly cost × months.
$y = 20 + 25x$
## Step3: Calculate 1-year cost (12 months) for Planet Fitness
Substitute $x=12$ into Planet Fitness equation.
$y = 50 + 29(12) = 50 + 348 = 398$
## Step4: Calculate 1-year cost (12 months) for LA Fitness
Substitute $x=12$ into LA Fitness equation.
$y = 20 + 25(12) = 20 + 300 = 320$
## Step5: Compare the two costs
$320 < 398$, so LA Fitness is cheaper.

# Answer:
a. System of equations:
Planet Fitness: $y = 29x + 50$
LA Fitness: $y = 25x + 20$
b. Mr. Saffer should choose LA Fitness. The total cost for 1 year is $320, which is $78 cheaper than Planet Fitness's $398.

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### Second Problem (McDonald's Ads)
# Explanation:
## Step1: Define total ad limit inequality
Total ads (Snapchat + Instagram) ≤ 16.
$x + y \leq 16$
## Step2: Define total cost inequality
Total cost ≥ $2400, with Snapchat ads at $400 and Instagram at $60.
$400x + 60y \geq 2400$ (simplifies to $20x + 3y \geq 120$)
## Step3: Add non-negativity constraints
Number of ads cannot be negative.
$x \geq 0, y \geq 0$
## Step4: Identify valid ad combination
Test $x=6, y=10$ against all inequalities:
1. $6 + 10 = 16 \leq 16$ (satisfies total ad limit)
2. $400(6) + 60(10) = 2400 + 600 = 3000 \geq 2400$ (satisfies cost requirement)
3. $6 \geq 0, 10 \geq 0$ (valid non-negative counts)

# Answer:
a. System of inequalities:
$$
\begin{cases}
x + y \leq 16 \
400x + 60y \geq 2400 \
x \geq 0 \
y \geq 0
\end{cases}
$$
(can be simplified to $\begin{cases}x + y \leq 16 \ 20x + 3y \geq 120 \ x \geq 0 \ y \geq 0\end{cases}$)
b. Graph:
1. Plot $x + y = 16$: connect (16,0) and (0,16), shade below the line.
2. Plot $20x + 3y = 120$: connect (6,0) and (0,40), shade above the line.
3. Shade only the region where all constraints overlap, label this area $S$.
c. One valid combination: 6 Snapchat ads and 10 Instagram ads. This uses the maximum 16 total ads, and the total cost is $3000, which meets the minimum $2400 requirement.

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QUESTION IMAGE

Question

Explanation:

First Problem (Gym Memberships)

Step1: Define Planet Fitness cost

Step2: Define LA Fitness cost

Step3: Calculate 1-year cost (12 months) for Planet Fitness

Step4: Calculate 1-year cost (12 months) for LA Fitness

Step5: Compare the two costs

Step1: Define total ad limit inequality

Step2: Define total cost inequality

Step3: Add non-negativity constraints

Step4: Identify valid ad combination

Answer:

Second Problem (McDonald's Ads)