Let's carefully analyze the problem statement and derive the equation step by step.
1. Entry Fee:
When a customer enters The Great Pumpkin Patch, they pay a fixed entry fee of [tex]$4. This fee is constant and does not depend on how many pounds of pumpkins they pick. Let's denote this entry fee by 4 dollars.
2. Cost per Pound:
Additionally, customers pay $[/tex]3 for each pound of pumpkins they pick. Let [tex]\( x \)[/tex] be the number of pounds of pumpkins they pick. Therefore, the total variable cost, which depends on the pounds of pumpkins picked, is [tex]\( 3x \)[/tex] dollars.
3. Total Cost:
The total cost [tex]\( y \)[/tex] consists of two parts: the fixed entry fee and the variable cost depending on the pounds of pumpkins picked. We can model this total cost as:
[tex]\[
y = 4 + 3x
\][/tex]
This equation combines the fixed fee and the variable fee to form the total cost [tex]\( y \)[/tex].
Now let's evaluate the given choices to see which one matches our derived equation:
- Choice A: [tex]\( y = 3(x + 4) \)[/tex]
[tex]\[
y = 3(x + 4) = 3x + 12
\][/tex]
This equation is incorrect because it suggests that the total cost depends differently on the pounds of pumpkins and adds an extra cost incorrectly.
- Choice B: [tex]\( y = 3x + 4 \)[/tex]
[tex]\[
y = 3x + 4
\][/tex]
This equation matches our derived equation.
- Choice C: [tex]\( y = 4x + 3 \)[/tex]
[tex]\[
y = 4x + 3
\][/tex]
This equation suggests a different relationship where [tex]\( 4x \)[/tex] is the variable cost, which is incorrect.
- Choice D: [tex]\( y = x(4 + 3) \)[/tex]
[tex]\[
y = x(4 + 3) = 7x
\][/tex]
This equation is incorrect as it improperly combines the fees.
Given this analysis, the correct equation that models the total cost [tex]\( y \)[/tex] for [tex]\( x \)[/tex] pounds of pumpkins is:
[tex]\[
\boxed{y = 3x + 4}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{B. } y = 3x + 4
\][/tex]
