Annalise makes rectangular patchwork quilts out of leftover fabric scraps. The first scrap she is using to make a quilt has a width of 2x feet and a length of 5x feet. When finished, the entire quilt will be 5 feet wider and 3 feet longer than the first fabric scrap used. Which of the following functions will give the area, f(x) , of the quilt in square feet?A. f(x)=10x^2+15 B. f(x)=10x^2+31x+15 C. f(x)=10x^2+25x+15 D. f(x)=10x^2+16x+15

1. Determine the dimensions of the finished quilt:
   • Width: \(2x + 5\)
   • Length: \(5x + 3\)
2. Calculate the area of the quilt:
   • Area = Width \(\times \) Length
   • Area = \(( 2x + 5) \times ( 5x + 3) \)
3. Expand the expression:
   \[( 2x + 5) ( 5x + 3) = 2x \cdot 5x + 2x \cdot 3 + 5 \cdot 5x + 5 \cdot 3\]
   \[= 10x^ 2 + 6x + 25x + 15\]
   \[= 10x^ 2 + 31x + 15\]
   So, the function \(f( x) \) that gives the area of the quilt in square feet is \(f( x) = 10x^ 2 + 31x + 15\).

Supplemental Knowledge

To solve this problem, we need to understand how to calculate the area of a rectangle and how to express it as a function. The area \(A\) of a rectangle is given by the formula:
\[A = \text { width} \times \text { length} \]
In this scenario, we have an initial rectangular piece of fabric with dimensions expressed in terms of \(x\). The width is \(2x\) feet and the length is \(5x\) feet. The entire quilt will be 5 feet wider and 3 feet longer than this initial piece.
Let's break down the steps:

1. Initial Dimensions:
   • Width: \(2x\)
   • Length: \(5x\)
2. Final Quilt Dimensions:
   • Width: \(2x + 5\)
   • Length: \(5x + 3\)

To find the area of the final quilt, we multiply these new dimensions together:
\[A = ( \text { width} ) \times ( \text { length} ) = ( 2x + 5) ( 5x + 3) \]
We can expand this product using the distributive property (FOIL method):
\[( 2x + 5) ( 5x + 3) = ( 2x \cdot 5x) + ( 2x \cdot 3) + ( 5 \cdot 5x) + ( 5 \cdot 3) \]
\[= 10x^ 2 + 6x + 25x + 15\]
\[= 10x^ 2 + 31x + 15\]

Knowledge in Action

Annalise's project can serve as an illustration of algebra applied in crafting. When planning quilts, Annalise uses algebraic expressions to calculate how much fabric she'll require in each part. By understanding polynomial functions to calculate areas more efficiently and plan materials more precisely each time - her quilts always end up the perfect size!

If you love crafting or want to enhance your understanding of algebraic functions like Annalise does with her quilts, check out UpStudy’s Algebra Functions Calculator! This tool can help you master polynomial expressions and other algebraic concepts with ease.
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