Which piecewise function models the cost of x pounds of trout? Karen owns a seafood restaurant. She orders trout from an online retailer. Each pound of trout costs $32, and the com... Which piecewise function models the cost of x pounds of trout? Karen owns a seafood restaurant. She orders trout from an online retailer. Each pound of trout costs $32, and the company charges a $4 fee for shipping the order. However, if Karen orders 10 or more pounds, the trout costs only $25 per pound, but the shipping fee is $9.
Understand the Problem
The question asks to identify the correct piecewise function that models the cost of trout based on the quantity ordered. The cost is $32 per pound with a $4 shipping fee for orders less than 10 pounds. For orders of 10 pounds or more, the cost is $25 per pound with a $9 shipping fee. We must choose the function that accurately represents these conditions.
Answer
C. $f(x) = \begin{cases} 32x + 4, & 0 < x < 10 \\ 25x + 9, & x \ge 10 \end{cases}$
Answer for screen readers
C. $f(x) = \begin{cases} 32x + 4, & 0 < x < 10 \ 25x + 9, & x \ge 10 \end{cases}$
Steps to Solve
- Identify costs for orders less than 10 pounds
For ( x < 10 ), the cost is $32 per pound plus a $4 shipping fee. Therefore, the cost function is $32x + 4$.
- Identify costs for orders of 10 pounds or more
For ( x \ge 10 ), the cost is $25 per pound plus a $9 shipping fee. Therefore, the cost function is $25x + 9$.
- Construct the piecewise function
Combine the two cost functions based on the quantity ordered:
$$ f(x) = \begin{cases} 32x + 4, & 0 < x < 10 \ 25x + 9, & x \ge 10 \end{cases} $$
- Compare with the given options
The correct piecewise function representation is choice C.
C. $f(x) = \begin{cases} 32x + 4, & 0 < x < 10 \ 25x + 9, & x \ge 10 \end{cases}$
More Information
Piecewise functions are very useful to represent different scenarios with different conditions. In this particular problem, the conditions depend on the quantity of trout purchased.
Tips
A common mistake might be to mix up the cost and shipping fee for the different order sizes, or to incorrectly assign the pound thresholds. Careful reading and labeling each cost can help avoid such mistakes.
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