A local screen-printing company sells T-shirts at various prices, in dollars, depending upon how many T-shirts are bought at the time. The piecewise function below represents their pricing structure, where [tex]$n$[/tex] is the number of [tex]$T$[/tex]-shirts purchased.
[tex]y=\left\{\begin{array}{l} 13 n, 0\ \textless \ n \leq 20 \\ 11 n, 20\ \textless \ n \leq 50 \\ 9 n, n\ \textgreater \ 50 \end{array}\right.[/tex]
How much did the volleyball team spend on shirts?
Answer (2)
Determine that the relevant part of the piecewise function is y = 13 n .
Substitute n = 20 into the equation: y = 13 × 20 .
Calculate the total cost: y = 260 .
The volleyball team spent 260 dollars.
Explanation
Analyze the problem The problem provides a piecewise function that determines the cost of T-shirts based on the number purchased. The volleyball team bought 20 shirts, so we need to determine which piece of the function applies when n = 20 .
Determine the applicable equation The piecewise function is defined as:
y = { 13 n , 050
Since the volleyball team purchased 20 shirts, the condition 0 < n ≤ 20 applies. Therefore, we use the equation y = 13 n to calculate the total cost.
Calculate the total cost Substitute n = 20 into the equation y = 13 n :
y = 13 × 20
y = 260
State the final answer The volleyball team spent $260 on shirts.
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, cell phone plans often have different rates for data usage depending on how much data you've used in a month. The first few gigabytes might be cheaper, but once you exceed a certain limit, the price per gigabyte increases. Similarly, electricity bills often have tiered pricing, where the cost per kilowatt-hour changes as your usage increases. Understanding piecewise functions helps you analyze and predict costs in these scenarios.
The volleyball team spent $260 on T-shirts by purchasing 20 shirts at a price of $13 each, as determined by the piecewise pricing function. We used the equation y = 13 n since n = 20 falls within the specified range. After calculating, the total expense was $260.
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