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Karen owns a seafood restaurant and orders trout from an online retailer. Each pound of trout costs [tex]$30, and the company charges a $[/tex]2 fee for shipping. If Karen orders 10 or more pounds, the trout costs [tex]$24 per pound, but the shipping fee is $[/tex]6.

Which piecewise function models the cost of [tex]\( x \)[/tex] pounds of trout?

A. [tex]\( f(x)=\left\{\begin{array}{l}24x + 6, \quad 0 \ \textless \ x \leq 10 \\ 30x + 2, \quad x \ \textgreater \ 10\end{array}\right. \)[/tex]

B. [tex]\( f(x)=\left\{\begin{array}{l}24x + 6, \quad 0 \ \textless \ x \ \textless \ 10 \\ 30x + 2, \quad x \geq 10\end{array}\right. \)[/tex]

C. [tex]\( f(x)=\left\{\begin{array}{ll}30x + 2, & 0 \ \textless \ x \ \textless \ 10 \\ 24x + 6, & x \geq 10\end{array}\right. \)[/tex]

D. [tex]\( f(x)=\left\{\begin{array}{l}30x + 2, \quad 0 \ \textless \ x \leq 10 \\ 24x + 6, \quad x \ \textgreater \ 10\end{array}\right. \)[/tex]



Answer :

GinnyAnswer
To determine the correct piecewise function modeling the cost of [tex]\( x \)[/tex] pounds of trout for Karen, let's analyze the conditions given in the problem and match them with the provided options.

1. For orders between 0 and 10 pounds (excluding exactly 10 pounds):
- The cost is \[tex]$30 per pound. - The shipping fee is \$[/tex]2.
- So, for [tex]\( 0 < x < 10 \)[/tex], the cost function would be [tex]\( 30x + 2 \)[/tex].

2. For orders of 10 or more pounds:
- The cost becomes \[tex]$24 per pound starting at 10 pounds. - The shipping fee changes to \$[/tex]6.
- So, for [tex]\( x \geq 10 \)[/tex], the cost function would be [tex]\( 24x + 6 \)[/tex].

Given these conditions, it's clear that:
- For [tex]\( 0 < x < 10 \)[/tex], the cost is [tex]\( 30x + 2 \)[/tex].
- For [tex]\( x \geq 10 \)[/tex], the cost is [tex]\( 24x + 6 \)[/tex].

Now, we compare these expressions to each given option:

A. [tex]\( f(x)=\left\{\begin{array}{l}24 x+6,\ 010\end{array}\right. \)[/tex]
- For [tex]\( 0 < x \leq 10 \)[/tex], it incorrectly assigns the cost as [tex]\( 24x + 6 \)[/tex]. This is incorrect because for [tex]\( 0 < x \leq 10 \)[/tex], the cost should be [tex]\( 30x + 2 \)[/tex].

B. [tex]\( f(x)=\left\{\begin{array}{l}24 x+6,\ 0 - For [tex]\( 0 < x < 10 \)[/tex], it incorrectly assigns the cost as [tex]\( 24x + 6 \)[/tex]. This is incorrect for the same reason as option A.

C. [tex]\( f(x)=\left\{\begin{array}{ll}30 x+2, & 0 - For [tex]\( 0 < x < 10 \)[/tex], the cost is correctly assigned as [tex]\( 30x + 2 \)[/tex].
- For [tex]\( x \geq 10 \)[/tex], the cost is correctly assigned as [tex]\( 24x + 6 \)[/tex]. This matches the conditions exactly.

D. [tex]\( f(x)=\left\{\begin{array}{l}30 x+2,\ 010\end{array}\right. \)[/tex]
- For [tex]\( 0 < x \leq 10 \)[/tex], the cost is correctly assigned as [tex]\( 30x + 2 \)[/tex] but includes the condition [tex]\( x = 10 \)[/tex]. This is incorrect because at exactly 10 pounds, the cost changes based on the new pricing and shipping fees.

Thus, the piecewise function [tex]\( f(x) \)[/tex] that correctly models the cost of [tex]\( x \)[/tex] pounds of trout for Karen is:

C. [tex]\( f(x)=\left\{\begin{array}{ll}30 x+2, & 0

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