The Correct Answer and Explanation is:
The correct answer is B. f(x) = 10x² + 31x + 15.
Here is a detailed explanation of how to arrive at this answer.
The problem asks for a function, f(x), that represents the area of the finished quilt. To find the area of any rectangle, we use the formula: Area = Length × Width. Therefore, our first step is to determine the expressions for the final length and width of the quilt.
First, we identify the dimensions of the initial fabric scrap. The problem states it has a width of 2x feet and a length of 5x feet.
Next, we calculate the dimensions of the finished quilt. The problem describes how the quilt’s final size relates to the initial scrap. It will be “5 feet wider” and “3 feet longer” than the first scrap.
To find the final width, we add 5 feet to the initial width:
Final Width = (Initial Width) + 5
Final Width = (2x + 5) feet
To find the final length, we add 3 feet to the initial length:
Final Length = (Initial Length) + 3
Final Length = (5x + 3) feet
Now that we have the expressions for the final length and width, we can calculate the area, f(x), by multiplying them together:
f(x) = (Final Length) × (Final Width)
f(x) = (5x + 3)(2x + 5)
To simplify this expression, we multiply the two binomials using the FOIL method (First, Outer, Inner, Last):
- First: Multiply the first terms of each binomial: (5x) × (2x) = 10x²
- Outer: Multiply the two outer terms: (5x) × (5) = 25x
- Inner: Multiply the two inner terms: (3) × (2x) = 6x
- Last: Multiply the last terms of each binomial: (3) × (5) = 15
Now, we combine all these products:
f(x) = 10x² + 25x + 6x + 15
Finally, we combine the like terms (25x and 6x):
f(x) = 10x² + 31x + 15
This resulting function matches option B.
