Answer :
To determine the appropriate piecewise function for modeling the cost of [tex]\(x\)[/tex] pounds of trout where each pound costs [tex]$\$[/tex] 30[tex]$ with a $[/tex]\[tex]$ 2$[/tex] shipping fee for orders less than 10 pounds, and [tex]$\$[/tex] 24[tex]$ per pound with a $[/tex]\[tex]$ 6$[/tex] shipping fee for orders of 10 or more pounds, we analyze the conditions and construct a piecewise function accordingly.
Here's how we proceed:
1. Condition for [tex]\(0 < x < 10\)[/tex]:
For an order of trout that weighs more than 0 pounds but less than 10 pounds:
- The cost per pound of trout is [tex]$\$[/tex] 30[tex]$. - The shipping fee is a fixed $[/tex]\[tex]$ 2$[/tex].
Thus, the cost function in this range can be represented as:
[tex]\[ f(x) = 30x + 2 \quad \text{for} \ 0 < x < 10 \][/tex]
2. Condition for [tex]\(x \geq 10\)[/tex]:
For an order of 10 pounds or more:
- The cost per pound of trout is reduced to [tex]$\$[/tex] 24[tex]$. - The shipping fee increases to $[/tex]\[tex]$ 6$[/tex].
Thus, the cost function in this range can be represented as:
[tex]\[ f(x) = 24x + 6 \quad \text{for} \ x \geq 10 \][/tex]
After constructing the two parts of the piecewise function, we need to choose the correct option among the given ones. The piecewise function should correctly capture the cost for both intervals:
[tex]\[ f(x) = \begin{cases} 30x + 2 & \text{if} \ 0 < x < 10 \\ 24x + 6 & \text{if} \ x \geq 10 \end{cases} \][/tex]
By comparing this with the given options:
- Option A:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x \leq 10 \\ 24x + 6 & x > 10 \end{cases} \][/tex]
This is incorrect because it incorrectly includes [tex]\( x = 10 \)[/tex] in the first condition.
- Option B:
[tex]\[ f(x) = \begin{cases} 24x + 6 & 0 < x < 10 \\ 30x + 2 & x \geq 10 \end{cases} \][/tex]
This is incorrect because it reverses the cost and shipping fee intervals.
- Option C:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x < 10 \\ 24x + 6 & x \geq 10 \end{cases} \][/tex]
This is correct since it matches our derived function perfectly.
- Option D:
[tex]\[ f(x) = \begin{cases} 24x + 6 & 0 < x \leq 10 \\ 30x + 2 & x > 10 \end{cases} \][/tex]
This is incorrect and similar to Option A but reversed.
Thus, the most accurate and correct piecewise function model is given by Option C:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x < 10 \\ 24x + 6 & x \geq 10 \end{cases} \][/tex]
Here's how we proceed:
1. Condition for [tex]\(0 < x < 10\)[/tex]:
For an order of trout that weighs more than 0 pounds but less than 10 pounds:
- The cost per pound of trout is [tex]$\$[/tex] 30[tex]$. - The shipping fee is a fixed $[/tex]\[tex]$ 2$[/tex].
Thus, the cost function in this range can be represented as:
[tex]\[ f(x) = 30x + 2 \quad \text{for} \ 0 < x < 10 \][/tex]
2. Condition for [tex]\(x \geq 10\)[/tex]:
For an order of 10 pounds or more:
- The cost per pound of trout is reduced to [tex]$\$[/tex] 24[tex]$. - The shipping fee increases to $[/tex]\[tex]$ 6$[/tex].
Thus, the cost function in this range can be represented as:
[tex]\[ f(x) = 24x + 6 \quad \text{for} \ x \geq 10 \][/tex]
After constructing the two parts of the piecewise function, we need to choose the correct option among the given ones. The piecewise function should correctly capture the cost for both intervals:
[tex]\[ f(x) = \begin{cases} 30x + 2 & \text{if} \ 0 < x < 10 \\ 24x + 6 & \text{if} \ x \geq 10 \end{cases} \][/tex]
By comparing this with the given options:
- Option A:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x \leq 10 \\ 24x + 6 & x > 10 \end{cases} \][/tex]
This is incorrect because it incorrectly includes [tex]\( x = 10 \)[/tex] in the first condition.
- Option B:
[tex]\[ f(x) = \begin{cases} 24x + 6 & 0 < x < 10 \\ 30x + 2 & x \geq 10 \end{cases} \][/tex]
This is incorrect because it reverses the cost and shipping fee intervals.
- Option C:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x < 10 \\ 24x + 6 & x \geq 10 \end{cases} \][/tex]
This is correct since it matches our derived function perfectly.
- Option D:
[tex]\[ f(x) = \begin{cases} 24x + 6 & 0 < x \leq 10 \\ 30x + 2 & x > 10 \end{cases} \][/tex]
This is incorrect and similar to Option A but reversed.
Thus, the most accurate and correct piecewise function model is given by Option C:
[tex]\[ f(x) = \begin{cases} 30x + 2 & 0 < x < 10 \\ 24x + 6 & x \geq 10 \end{cases} \][/tex]
