Let's use a step-by-step approach to determine the area of the new pumpkin patch.
Step 1: Identify the initial dimensions of the pumpkin patch:
- The initial width is [tex]\( 40 \)[/tex] meters.
- The initial length is [tex]\( 60 \)[/tex] meters.
Step 2: Determine the increase in dimensions:
- The width increases by [tex]\( 3x \)[/tex] meters.
- The length increases by [tex]\( 5x \)[/tex] meters.
Step 3: Calculate the new dimensions:
- The new width will be [tex]\( 40 + 3x \)[/tex] meters.
- The new length will be [tex]\( 60 + 5x \)[/tex] meters.
Step 4: Define the formula for the area of a rectangle:
- Area = width × length.
Step 5: Substitute the new dimensions into the area formula:
[tex]\[ \text{Area}_{\text{new}} = (40 + 3x)(60 + 5x) \][/tex]
Step 6: Use the distributive property (expand the product of the binomials):
[tex]\[ \text{Area}_{\text{new}} = 40 \cdot 60 + 40 \cdot 5x + 3x \cdot 60 + 3x \cdot 5x \][/tex]
[tex]\[ \text{Area}_{\text{new}} = 2400 + 200x + 180x + 15x^2 \][/tex]
[tex]\[ \text{Area}_{\text{new}} = 2400 + 380x + 15x^2 \][/tex]
Step 7: Rewrite the expression in standard polynomial form:
[tex]\[ \text{Area}_{\text{new}} = 15x^2 + 380x + 2400 \][/tex]
Thus, the function that gives the area of the new pumpkin patch in square meters is:
[tex]\[ f(x) = 15x^2 + 380x + 2400 \][/tex]
Comparing this with the given options, the correct answer is:
[tex]\[ \boxed{f(x) = 15x^2 + 380x + 2400} \][/tex]
The correct answer is [tex]\( C \)[/tex].
