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Select the correct answer.

Joe wants to enlarge the rectangular pumpkin patch on his farm. The current patch is 40 meters wide and 60 meters long. The new patch will be [tex]\(3x\)[/tex] meters wider and [tex]\(5x\)[/tex] meters longer.

Which function gives the area of the new pumpkin patch in square meters?

A. [tex]\(f(x)=15x^2\)[/tex]
B. [tex]\(f(x)=15x^2 + 2,400\)[/tex]
C. [tex]\(f(x)=15x^2 + 380x + 2,400\)[/tex]
D. [tex]\(f(x)=15x^2 + 420x + 2,400\)[/tex]



Answer :

GinnyAnswer
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].

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