Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

The length of a rectangle is represented by the function [tex]\(L(x)=5x\)[/tex]. The width of that same rectangle is represented by the function [tex]\(W(x)=2x^2-4x+13\)[/tex]. Which of the following shows the area of the rectangle in terms of [tex]\(x\)[/tex]?

A. [tex]\(A(x)=10x^3-4x+13\)[/tex]

B. [tex]\(A(x)=10x^3-20x^2+65x\)[/tex]

C. [tex]\(A(x)=2x^3+4x+49\)[/tex]

D. [tex]\(A(x)=2x^2-9x+13\)[/tex]


Sagot :

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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.