Question

an engineer is designing a building and wants rectangular spaces to maintain a ratio of 2 to 3. for a measurement value x, the length and width of the rectangular spaces can be represented by the fun
length: $f(x) = 3x$
width: $g(x) = 2x$
which function represents the area for any rectangular space in the building?
$\bigcirc\\ 6x^2$
$\bigcirc\\ 6x$
$\bigcirc\\ 3x^2$
$\bigcirc\\ 10x$

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by the product of its length and width, i.e., \( A=\text{length} \times \text{width} \).

Step2: Substitute length and width functions

Given length \( f(x) = 3x \) and width \( g(x)=2x \), we substitute these into the area formula: \( A(x)=f(x)\times g(x)=(3x)\times(2x) \).

Step3: Simplify the product

Using the rule of exponents \( a^m\times a^n=a^{m + n} \) and the coefficient multiplication \( 3\times2 = 6 \), we get \( (3x)\times(2x)=6x^{1 + 1}=6x^{2} \).

Answer:

\( 6x^{2} \) (the option corresponding to \( 6x^2 \))