Question

1. a new cell phone company offers two monthly plans

plan a: a flat fee of $25 per month plus $0.10 per text message.
plan b: a flat fee of $40 per month plus $0.050 per text message.
john wants to find out after how many text messages both plans cost the same amount and which plan is cheaper if they send more than that number of texts. verify the answer using desmos graph.

1. two chess clubs offer different monthly payment options.

club a: a $20 sign - up fee plus 415 per class.
club b: a $50 sign - up fee plus $10 per class.
carla wants to know
a. how many classes she must take for both clubs to cost the same.
b. which club is cheaper if she takes 12 classes in a month.
verify your answer using the desmos graph.

Explanation:

Step1: Set up equations for Plan A and Plan B

Let \(x\) be the number of text messages.
For Plan A: \(y = 25+0.1x\)
For Plan B: \(y = 40 + 0.05x\)

Step2: Solve for \(x\) when \(y\) is equal

Set \(25+0.1x=40 + 0.05x\)
\(0.1x-0.05x=40 - 25\)
\(0.05x=15\)
\(x=\frac{15}{0.05}=300\)

Step3: Analyze which plan is cheaper for \(x>300\)

Take \(x = 400\)
Plan A: \(y=25+0.1\times400=25 + 40=65\)
Plan B: \(y=40+0.05\times400=40 + 20=60\)
Since \(60<65\), Plan B is cheaper for \(x > 300\)

Step4: Set up equations for Club A and Club B (problem 2)

Let \(x\) be the number of classes
For Club A: \(y=20 + 15x\)
For Club B: \(y=50+10x\)

Step5: Solve for \(x\) when \(y\) is equal (problem 2a)

Set \(20+15x=50+10x\)
\(15x-10x=50 - 20\)
\(5x=30\)
\(x = 6\)

Step6: Calculate cost for \(x = 12\) (problem 2b)

Club A: \(y=20+15\times12=20+180=200\)
Club B: \(y=50+10\times12=50+120=170\)
Since \(170<200\), Club B is cheaper for \(x = 12\)

Answer:

1. After \(300\) text messages, both plans cost the same. Plan B is cheaper if more than \(300\) texts are sent.
2. a. \(6\) classes. b. Club B is cheaper.