Question

Ella is deciding between two different movie streaming sites to subscribe to. Plan A costs $20 per month plus $1 per movie watched. Plan B costs $10 per month plus $3 per movie watched. Let AA represent the monthly cost of Plan A if Ella watches xx per month, and let BB represent the monthly cost of Plan B if Ella watches xx movies per month. Graph each function and determine the interval of movies watched, x, commax, for which Plan A is cheaper than Plan B.

Answer 1

Let's start by defining the cost functions for both Plan A and Plan B in terms of the number of movies watched, \(x\).

1. Plan A: \[ A(x) = 20 + 1x = 20 + x \]

2. Plan B: \[ B(x) = 10 + 3x \]

Next, we need to find the interval for which Plan A is cheaper than Plan B. This can be expressed as an inequality: \[ A(x) < B(x) \]

Substituting the functions we defined earlier, we get: \[ 20 + x < 10 + 3x \]

Now, we'll solve for \(x\):

1. Subtract \(x\) from both sides: \[ 20 < 10 + 2x \]

2. Subtract 10 from both sides: \[ 10 < 2x \]

3. Divide both sides by 2: \[ 5 < x \quad \text{or} \quad x > 5 \]

Thus, Plan A is cheaper than Plan B when the number of movies watched, \(x\), is greater than 5.

Graphing the Functions

To graph the functions \(A(x)\) and \(B(x)\):

1. Plan A (\(A(x) = 20 + x\)):

   • When \(x = 0\), \(A(0) = 20\)
   • When \(x = 5\), \(A(5) = 25\)
   • When \(x = 10\), \(A(10) = 30\)

2. Plan B (\(B(x) = 10 + 3x\)):

   • When \(x = 0\), \(B(0) = 10\)
   • When \(x = 5\), \(B(5) = 25\)
   • When \(x = 10\), \(B(10) = 40\)

Plotting the Points

• For Plan A (Linear with a slope of 1 and y-intercept 20):

  • Points: (0, 20), (5, 25), (10, 30)

• For Plan B (Linear with a slope of 3 and y-intercept 10):

  • Points: (0, 10), (5, 25), (10, 40)

Conclusion

The interval for the number of movies watched \(x\) such that Plan A is cheaper than Plan B is: \[ \text{For } x > 5 \]

In summary, Ella should choose Plan A when she watches more than 5 movies in a month.