To determine the area of a rectangle when given the length and width as functions of [tex]\( x \)[/tex], we simply multiply these functions together.
Given:
- [tex]\( L(x) = 5x \)[/tex] (Length as a function of [tex]\( x \)[/tex])
- [tex]\( W(x) = 2x^2 - 4x + 13 \)[/tex] (Width as a function of [tex]\( x \)[/tex])
The area [tex]\( A(x) \)[/tex] of the rectangle as a function of [tex]\( x \)[/tex] is given by:
[tex]\[ A(x) = L(x) \times W(x) \][/tex]
Step-by-Step Solution:
1. Substitute the expressions for [tex]\( L(x) \)[/tex] and [tex]\( W(x) \)[/tex] into the area function:
[tex]\[ A(x) = (5x) \times (2x^2 - 4x + 13) \][/tex]
2. Distribute [tex]\( 5x \)[/tex] to each term inside the parentheses:
[tex]\[ A(x) = 5x \times 2x^2 + 5x \times (-4x) + 5x \times 13 \][/tex]
3. Multiply the terms:
[tex]\[ 5x \times 2x^2 = 10x^3 \][/tex]
[tex]\[ 5x \times (-4x) = -20x^2 \][/tex]
[tex]\[ 5x \times 13 = 65x \][/tex]
4. Combine the results:
[tex]\[ A(x) = 10x^3 - 20x^2 + 65x \][/tex]
Thus, the area of the rectangle in terms of [tex]\( x \)[/tex] is:
[tex]\[ 10x^3 - 20x^2 + 65x \][/tex]
So, the correct answer is:
[tex]\[ W(x) = 10x^3 - 20x^2 + 65x \][/tex]
