College

The area of a rectangle, [tex]A = l \cdot w[/tex], is represented by the expression [tex]24 x^6 y^{15}[/tex]. Which could be the dimensions of the rectangle?

A. [tex]2 x^5 y^8[/tex] and [tex]12 x y^7[/tex]

B. [tex]6 x^2 y^3[/tex] and [tex]4 x^3 y^5[/tex]

C. [tex]10 x^6 y^{15}[/tex] and [tex]14 x^6 y^{15}[/tex]

D. [tex]9 x^4 y^{11}[/tex] and [tex]12 x^2 y^4[/tex]

Answer :

We are given that the area of a rectangle is expressed as
[tex]$$24x^6y^{15},$$[/tex]
and we need to determine which set of dimensions (length and width) would give this area when multiplied.

Let the dimensions be [tex]$L$[/tex] and [tex]$W$[/tex]. Their product must equal the area:
[tex]$$L \times W = 24x^6y^{15}.$$[/tex]

Now, consider the dimensions:
[tex]$$L = 2x^5y^8 \quad \text{and} \quad W = 12xy^7.$$[/tex]

We multiply these to check if they produce the given area:
[tex]\[
\begin{aligned}
(2x^5y^8) \times (12xy^7) &= (2 \times 12) \times (x^5 \times x) \times (y^8 \times y^7) \\
&= 24 \times x^{5+1} \times y^{8+7} \\
&= 24x^6y^{15}.
\end{aligned}
\][/tex]

Since the product equals the given area, the dimensions are correct.

Thus, the dimensions of the rectangle are
[tex]$$\boxed{2x^5y^8 \text{ and } 12xy^7.}$$[/tex]