Sure! Let's solve the problem step-by-step:
1. Identify the Initial Dimensions:
- The initial width of the pumpkin patch is 40 meters.
- The initial length of the pumpkin patch is 60 meters.
2. Determine the New Dimensions:
- The new width will be [tex]\(40 + 3x\)[/tex] meters.
- The new length will be [tex]\(60 + 5x\)[/tex] meters.
Here, [tex]\(x\)[/tex] is a variable that represents how many meters the width and length are increased.
3. Express the Area of the New Pumpkin Patch:
- The area of a rectangle is calculated as width multiplied by length.
- So, the area [tex]\(A\)[/tex] of the new pumpkin patch will be:
[tex]\[
A = (40 + 3x) \times (60 + 5x)
\][/tex]
4. Expand the Expression:
- To find the expanded form, we need to distribute the terms:
[tex]\[
(40 + 3x) \times (60 + 5x)
\][/tex]
- Apply the distributive property (FOIL method):
[tex]\[
(40 \times 60) + (40 \times 5x) + (3x \times 60) + (3x \times 5x)
\][/tex]
- First Term: [tex]\(40 \times 60 = 2400\)[/tex]
- Outer Term: [tex]\(40 \times 5x = 200x\)[/tex]
- Inner Term: [tex]\(3x \times 60 = 180x\)[/tex]
- Last Term: [tex]\(3x \times 5x = 15x^2\)[/tex]
- Combine all these terms:
[tex]\[
2400 + 200x + 180x + 15x^2
\][/tex]
- Simplify by combining like terms:
[tex]\[
2400 + (200x + 180x) + 15x^2 = 2400 + 380x + 15x^2
\][/tex]
5. Conclusion:
- The function that represents the area of the new pumpkin patch in square meters is:
[tex]\[
f(x) = 15x^2 + 380x + 2400
\][/tex]
- Therefore, the correct answer is:
[tex]\[
\boxed{A. \ f(x) = 15 x^2 + 380 x + 2400}
\][/tex]
