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Karen owns a seafood restaurant. She orders trout from an online retailer. Each pound of trout costs [tex]$28$[/tex], and the company charges a [tex]$4$[/tex] fee for shipping the order. However, if Karen orders 10 or more pounds, the trout costs only [tex]$22$[/tex] per pound, but the shipping fee is [tex]$8$[/tex].

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

A. [tex] f(x)=\left\{ \begin{array}{l} 28x+4, \; 0 \ \textless \ x \ \textless \ 10 \\ 22x+8, \; x \geq 10 \end{array} \right. [/tex]

B. [tex] f(x)=\left\{ \begin{array}{ll} 28x+4, & \; 0 \ \textless \ x \leq 10 \\ 22x+8, & \; x \ \textgreater \ 10 \end{array} \right. [/tex]

C. [tex] f(x)=\left\{ \begin{array}{l} 22x+8, \; 0 \ \textless \ x \leq 10 \\ 28x+4, \; x \ \textgreater \ 10 \end{array} \right. [/tex]



Answer :

GinnyAnswer
To determine which piecewise function best models the cost [tex]\( f(x) \)[/tex] of ordering [tex]\( x \)[/tex] pounds of trout from Karen's online retailer, let's carefully analyze the problem's conditions:

1. For orders less than 10 pounds ([tex]\( 0 < x < 10 \)[/tex]):
- Each pound of trout costs \[tex]$28. - There is a shipping fee of \$[/tex]4.

Therefore, the cost function over this interval is:
[tex]\[ f(x) = 28x + 4 \][/tex]

2. For orders of 10 pounds or more ([tex]\( x \geq 10 \)[/tex]):
- Each pound of trout costs \[tex]$22. - There is a shipping fee of \$[/tex]8.

Thus, the cost function over this interval is:
[tex]\[ f(x) = 22x + 8 \][/tex]

Given these conditions, the appropriate piecewise function for the cost [tex]\( f(x) \)[/tex] of ordering [tex]\( x \)[/tex] pounds of trout is:

[tex]\[ f(x) = \begin{cases} 28x + 4, & \text{if } 0 < x < 10 \\ 22x + 8, & \text{if } x \geq 10 \end{cases} \][/tex]

So, the correct piecewise function is:

A. [tex]\( f(x) = \left\{ \begin{array}{ll} 28x + 4, & 0 < x < 10 \\ 22x + 8, & x \geq 10 \end{array} \right. \)[/tex]

Thus, the answer is:
[tex]\[ \boxed{1} \][/tex]