QuestionQuestion 1 (6 points)
1. A college official is purchasing some spirit gear to give to campus visitors. She can purchase Yeti mugs with the college logo engraved on them for per mug for the first 24 mugs. For a purchase of 25 or more mugs, the price per mug is reduced by for all the mugs over 24.
a. Write a piecewise function to represent the cost, to the college when mugs are purchased. ( 6 points)
Pargraph
Studdy Solution
STEP 1
1. The cost of each Yeti mug is \$38 for the first 24 mugs.
2. For purchases of 25 or more mugs, the price per mug is reduced by \$8 for each mug over 24.
3. We need to write a piecewise function to represent the total cost \( C(x) \) for purchasing \( x \) mugs.
STEP 2
1. Define the cost function for purchasing up to 24 mugs.
2. Define the cost function for purchasing 25 or more mugs.
3. Combine the two expressions into a piecewise function.
STEP 3
For , the cost function is straightforward since each mug costs \$38. Therefore, the cost for \( x \) mugs is:
STEP 4
For , the first 24 mugs cost \$38 each, and the remaining mugs (those over 24) cost \$30 each (since \$38 - \$8 = \$30). Therefore, the cost function is:
STEP 5
Combine the expressions from Step 1 and Step 2 into a piecewise function:
\[
C(x) =
\begin{cases}
38x, & \text{if } x \leq 24 \\
38 \times 24 + 30 \times (x - 24), & \text{if } x \geq 25
\end{cases}$
The piecewise function representing the cost for purchasing mugs is:
\[
C(x) =
\begin{cases}
38x, & \text{if } x \leq 24 \\
912 + 30(x - 24), & \text{if } x \geq 25
\end{cases}$
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