A cell phone company charges by the minute (and partial minute) for making phone calls. Arionna's plan includes 300 minutes in the $20 monthly base cost. If she uses more than 300 minutes in a month, there is a $5 overage fee and an additional charge of $0.25 per minute. Which graph represents the monthly cost, $y$, in dollars for making $x$ minutes of calls?
Answer (2)
The monthly cost y is $20 for the first 300 minutes. After 300 minutes, there is a $5 overage fee and an additional charge of $0.25 per minute. The function representing this is: y = 20 if 0
\[y = 0.25x - 50\] if x > 300 . T h e g r a p hi s ah or i zo n t a ll in e a t y=20$ for 0 ≤ x ≤ 300 , then a line with slope 0.25 starting at ( 300 , 25 ) .
Explanation
Understanding the Problem Let's analyze the problem. We have a cell phone plan with a base cost of $20 for the first 300 minutes. If the usage exceeds 300 minutes, there is a $5 overage fee and an additional charge of $0.25 per minute. We need to determine the function representing the monthly cost y in dollars for making x minutes of calls, and relate it to the correct graph.
Defining the Cost Function For 0
\[0 \]For x > 300$, the cost is the base cost plus the overage fee plus the charge for the additional minutes, which can be expressed as: y = 20 + 5 + 0.25 ( x − 300 )
Simplifying the Cost Function Simplify the expression for 300"> x > 300 :
y = 25 + 0.25 x − 0.25 ( 300 ) = 25 + 0.25 x − 75 = 0.25 x − 50 So, the function is defined as: 300 \end{cases}"> y = { 20 if 0 ≤ x ≤ 300 0.25 x − 50 if x > 300
Analyzing the Graph The graph should be a horizontal line at y = 20 for 0 ≤ x ≤ 300 . For 300"> x > 300 , the graph should be a line with a slope of 0.25. Let's find the value of y when x = 300 for the second part of the function: y = 0.25 ( 300 ) − 50 = 75 − 50 = 25 So, the line starts at the point (300, 25).
Conclusion The graph should be a horizontal line at y = 20 until x = 300 . After that, it should be a line with a slope of 0.25 starting at the point ( 300 , 25 ) .
Examples
Understanding cell phone billing plans can be tricky! This problem helps you understand how costs can be structured based on usage. Imagine you're comparing different phone plans. By expressing the cost as a function of minutes used, you can easily calculate and compare the costs for different usage scenarios. This kind of analysis is useful for budgeting and making informed decisions about which plan best suits your needs.
Arionna's monthly cost function is constant at $20 for the first 300 minutes and increases linearly at a rate of $0.25 per minute after that, starting at a cost of 25 w h e nh er u s a g ee x cee d s 300 min u t es . T h e g r a p h w i ll s h o w ah or i zo n t a ll in e a t y = 20 u n t i l x = 300$, then a line with a slope of 0.25 starting at the point ( 300 , 25 ) . Therefore, the cost representation corresponds to two segments based on her usage.
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