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 :

To solve the problem of finding the dimensions of a rectangle with an area of [tex]\(24x^6y^{15}\)[/tex], we need to check each set of given dimensions to see which pair, when multiplied, equals [tex]\(24x^6y^{15}\)[/tex].

Let's evaluate each option:

1. Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the lengths:
[tex]\[
(2x^5y^8) \times (12xy^7) = (2 \times 12)(x^5 \times x)(y^8 \times y^7) = 24x^{5+1}y^{8+7} = 24x^6y^{15}
\][/tex]
- This product matches the expression for the area, so these could be the dimensions of the rectangle.

2. Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the lengths:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = (6 \times 4)(x^2 \times x^3)(y^3 \times y^5) = 24x^{2+3}y^{3+5} = 24x^5y^8
\][/tex]
- The product here does not match the given area.

3. Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the lengths:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = (10 \times 14)(x^6 \times x^6)(y^{15} \times y^{15}) = 140x^{12}y^{30}
\][/tex]
- This product does not match the given area.

4. Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the lengths:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = (9 \times 12)(x^4 \times x^2)(y^{11} \times y^4) = 108x^{6}y^{15}
\][/tex]
- This product does not match the given area.

After evaluating all the options, the dimensions [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex] provide the correct product that equals the area of [tex]\(24x^6y^{15}\)[/tex].