Let's break down the problem step-by-step to create a mathematical model for the total cost [tex]\( y \)[/tex] when picking [tex]\( x \)[/tex] pounds of berries at Gooseberry Farm.
1. Entry Fee: The customers must pay an entry fee of \[tex]$3 to enter the farm. This fee is a fixed amount and does not depend on the number of pounds of berries picked. Thus, this part of the cost remains constant at \$[/tex]3.
2. Cost per Pound: The cost for each pound of berries is \[tex]$2.50. This cost will vary depending on the number of pounds of berries picked. So for \( x \) pounds, the cost will be \$[/tex]2.50 multiplied by [tex]\( x \)[/tex].
3. Total Cost: The total cost is the sum of the entry fee and the cost for the picked berries. This can be represented as:
[tex]\[
y = (\text{Entry Fee}) + (\text{Cost per Pound} \times \text{Number of Pounds})
\][/tex]
4. Inserting Values:
- Entry Fee: \[tex]$3
- Cost per Pound: \$[/tex]2.50
- Number of Pounds: [tex]\( x \)[/tex]
So the equation becomes:
[tex]\[
y = 3 + 2.5x
\][/tex]
To ensure we select the correct option, we can look at the provided answers:
A. [tex]\( y = 3x + 2.5 \)[/tex]
B. [tex]\( y = 2.5(x + 3) \)[/tex]
C. [tex]\( y = x(2.5 + 3) \)[/tex]
D. [tex]\( y = 2.5x + 3 \)[/tex]
Matching this with our equation [tex]\( y = 3 + 2.5x \)[/tex]:
- Option A is [tex]\( y = 3x + 2.5 \)[/tex]: This is incorrect because it incorrectly multiplies [tex]\( x \)[/tex] by 3 instead of 2.5.
- Option B is [tex]\( y = 2.5(x + 3) \)[/tex]: This simplifies to [tex]\( y = 2.5x + 7.5 \)[/tex], which is incorrect.
- Option C is [tex]\( y = x(2.5 + 3) \)[/tex]: This simplifies to [tex]\( y = x \times 5.5 \)[/tex], which is incorrect.
- Option D is [tex]\( y = 2.5x + 3 \)[/tex]: This is correct because it accurately represents adding the entry fee and the cost per pound times the number of pounds.
Thus, the correct option is [tex]\( \boxed{D} \)[/tex].
