Create a word problem about piecewise function.
Understand the Problem
The question asks for an example of a problem or word problem that would involve a piecewise function. We need to create a scenario where different rules or formulas apply depending on the input.
Answer
A cell phone company charges \$30 per month for up to 5 GB of data. For data usage exceeding 5 GB, they charge an additional \$10 per GB. If a customer uses 8 GB of data in a month, what is their total bill? The piecewise function is: $$ C(x) = \begin{cases} 30 & \text{if } 0 \le x \le 5 \\ 30 + 10(x - 5) & \text{if } x > 5 \end{cases} $$ The total bill is \$60.
Answer for screen readers
A cell phone company charges $30 per month for up to 5 GB of data. For data usage exceeding 5 GB, they charge an additional $10 per GB. If a customer uses 8 GB of data in a month, what is their total bill?
The piecewise function that represents the situation is: $$ C(x) = \begin{cases} 30 & \text{if } 0 \le x \le 5 \ 30 + 10(x - 5) & \text{if } x > 5 \end{cases} $$
The customer's total bill is $60.
Steps to Solve
- Define a scenario
Consider a cell phone plan where the cost varies based on the amount of data used. There's a fixed monthly fee for a certain amount of data, and then an additional charge for each gigabyte (GB) over that limit.
- Define the intervals
Let's say the plan includes 5 GB of data for a fixed cost of $30. For each additional GB over 5 GB, the charge is $10 per GB.
- Define the piecewise function
We can express this situation using a piecewise function, where $x$ represents the amount of data used in GB and $C(x)$ represents the total cost of the cell phone plan.
$$ C(x) = \begin{cases} 30 & \text{if } 0 \le x \le 5 \ 30 + 10(x - 5) & \text{if } x > 5 \end{cases} $$
- Describe an example word problem
Here's a word problem based on this piecewise function:
"A cell phone company charges $30 per month for up to 5 GB of data. For data usage exceeding 5 GB, they charge an additional $10 per GB. If a customer uses 8 GB of data in a month, what is their total bill?"
- Solve example word problem
Since the customer used 8 GB of data, which is more than 5 GB, we use the second part of the piecewise function:
$C(8) = 30 + 10(8 - 5) = 30 + 10(3) = 30 + 30 = 60$
The total bill is $60.
A cell phone company charges $30 per month for up to 5 GB of data. For data usage exceeding 5 GB, they charge an additional $10 per GB. If a customer uses 8 GB of data in a month, what is their total bill?
The piecewise function that represents the situation is: $$ C(x) = \begin{cases} 30 & \text{if } 0 \le x \le 5 \ 30 + 10(x - 5) & \text{if } x > 5 \end{cases} $$
The customer's total bill is $60.
More Information
Piecewise functions are very useful for modeling real-world situations where different conditions lead to different outcomes. Cell phone plans, taxi fares, and income tax brackets are common examples.
Tips
A common mistake is not correctly identifying the intervals for which each piece of the function applies. Also, some may forget to subtract the base amount of data (5 GB in this case) before multiplying by the overage charge. Choosing the incorrect piece of the function based on the input value is another common error.
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