Karen owns a seafood restaurant. She orders trout from an online retailer. Each pound of trout costs [tex]$\$32$[/tex], and the company charges a [tex]$\[tex]$4$[/tex][/tex] fee for shipping the order. However, if Karen orders 10 or more pounds, the trout costs only [tex]$\$25$[/tex] per pound, but the shipping fee is [tex]$\[tex]$9$[/tex][/tex].

Which piecewise function models the cost of [tex]$x$[/tex] pounds of trout?

A. [tex]f(x) = \left\{\begin{array}{ll}25x + 9, & 0 \ \textless \ x \ \textless \ 10 \\ 32x + 4, & x \geq 10\end{array}\right.[/tex]

B. [tex]f(x) = \left\{\begin{array}{l}32x + 4, \quad 0 \ \textless \ x \ \textless \ 10 \\ 25x + 9, \quad x \geq 10\end{array}\right.[/tex]

C. [tex]f(x) = \left\{\begin{array}{ll}25x + 9, & 0 \ \textless \ x \leq 10 \\ 32x + 4, & x \ \textgreater \ 10\end{array}\right.[/tex]



Answer :

To determine the correct piecewise function that models the cost [tex]\( f(x) \)[/tex] of [tex]\( x \)[/tex] pounds of trout, we need to consider both the per-pound cost and the shipping costs, based on the quantity Karen orders.

We'll break down the cost according to the amount she orders:

1. For orders less than 10 pounds:
- The cost per pound of trout is [tex]$32. - The shipping fee is $[/tex]4.
- Therefore, for [tex]\( 0 < x < 10 \)[/tex]:
[tex]\[ f(x) = 32x + 4 \][/tex]

2. For orders of 10 pounds or more:
- The cost per pound of trout drops to [tex]$25. - The shipping fee increases to $[/tex]9.
- Therefore, for [tex]\( x \geq 10 \)[/tex]:
[tex]\[ f(x) = 25x + 9 \][/tex]

Combining these two conditions, the piecewise function that models the cost [tex]\( f(x) \)[/tex] of trout can be written as:
[tex]\[ f(x) = \left\{ \begin{array}{ll} 32x + 4, & 0 < x < 10 \\ 25x + 9, & x \geq 10 \end{array} \right. \][/tex]

Now, comparing this result with the given options:

A. [tex]\( f(x) = \left\{ \begin{array}{ll} 25x + 9, & 0 < x < 10 \\ 32x + 4, & x \geq 10 \end{array} \right. \)[/tex]

B. [tex]\( f(x) = \left\{ \begin{array}{ll} 32x + 4, & 0 < x < 10 \\ 25x + 9, & x \geq 10 \end{array} \right. \)[/tex]

D. [tex]\( f(x) = \left\{ \begin{array}{ll} 25x + 9, & 0 < x \leq 10 \\ 32x + 4, & x > 10 \end{array} \right. \)[/tex]

Option B matches our derived piecewise function. Hence, the correct piecewise function is:

[tex]\[ f(x) = \left\{ \begin{array}{ll} 32x + 4, & 0 < x < 10 \\ 25x + 9, & x \geq 10 \end{array} \right. \][/tex]

Thus, the correct choice is B.