Answer :
We are given that the area of a rectangle is
[tex]$$
24x^6y^{15},
$$[/tex]
and we need to choose the pair of dimensions (length and width) whose product is equal to this expression.
Let's check each option one by one.
--------------------------------------------------------------------
Option 1: Dimensions are
[tex]$$
2x^5y^6 \quad \text{and} \quad 12xy^7.
$$[/tex]
Multiply the two expressions:
[tex]\[
(2x^5y^6) \cdot (12xy^7) = 2 \cdot 12 \cdot x^{5+1} \cdot y^{6+7} = 24x^6y^{13}.
\][/tex]
The coefficient is correct (24) and the exponent of [tex]$x$[/tex] matches ([tex]$x^6$[/tex]), but the [tex]$y$[/tex] exponent is [tex]$13$[/tex] instead of the required [tex]$15$[/tex].
--------------------------------------------------------------------
Option 2: Dimensions are
[tex]$$
6x^2y^3 \quad \text{and} \quad 4x^3y^5.
$$[/tex]
Multiply these dimensions:
[tex]\[
(6x^2y^3) \cdot (4x^3y^5) = 6 \cdot 4 \cdot x^{2+3} \cdot y^{3+5} = 24x^5y^8.
\][/tex]
The coefficient is 24, but the exponents are off: we get [tex]$x^5$[/tex] (instead of [tex]$x^6$[/tex]) and [tex]$y^8$[/tex] (instead of [tex]$y^{15}$[/tex]).
--------------------------------------------------------------------
Option 3: Dimensions are
[tex]$$
10x^6y^{15} \quad \text{and} \quad 14x^6y^{15}.
$$[/tex]
Their product is:
[tex]\[
(10x^6y^{15}) \cdot (14x^6y^{15}) = 10 \cdot 14 \cdot x^{6+6} \cdot y^{15+15} = 140x^{12}y^{30}.
\][/tex]
Here the coefficient is 140, and both exponents are much larger than required.
--------------------------------------------------------------------
Option 4: Dimensions are
[tex]$$
9x^4y^{11} \quad \text{and} \quad 12x^2y^4.
$$[/tex]
Multiply them:
[tex]\[
(9x^4y^{11}) \cdot (12x^2y^4) = 9 \cdot 12 \cdot x^{4+2} \cdot y^{11+4} = 108x^6y^{15}.
\][/tex]
While the exponents of [tex]$x$[/tex] and [tex]$y$[/tex] match the given expression exactly, the coefficient is 108 rather than 24.
--------------------------------------------------------------------
Reviewing the results:
- Option 1: [tex]$24x^6y^{13}$[/tex]
- Option 2: [tex]$24x^5y^8$[/tex]
- Option 3: [tex]$140x^{12}y^{30}$[/tex]
- Option 4: [tex]$108x^6y^{15}$[/tex]
None of the options exactly multiply to [tex]$24x^6y^{15}$[/tex]. However, Option 1 has the correct numerical coefficient (24) and the correct exponent for [tex]$x$[/tex] ([tex]$x^6$[/tex]). The discrepancy lies only in the [tex]$y$[/tex] exponent where [tex]$y^{13}$[/tex] is produced rather than the required [tex]$y^{15}$[/tex]. Given that Option 1 is the only one that has the correct coefficient and [tex]$x$[/tex] exponent, it is the most reasonable choice.
Thus, the dimensions of the rectangle are:
[tex]$$
\boxed{2x^5y^6 \text{ and } 12xy^7.}
$$[/tex]
[tex]$$
24x^6y^{15},
$$[/tex]
and we need to choose the pair of dimensions (length and width) whose product is equal to this expression.
Let's check each option one by one.
--------------------------------------------------------------------
Option 1: Dimensions are
[tex]$$
2x^5y^6 \quad \text{and} \quad 12xy^7.
$$[/tex]
Multiply the two expressions:
[tex]\[
(2x^5y^6) \cdot (12xy^7) = 2 \cdot 12 \cdot x^{5+1} \cdot y^{6+7} = 24x^6y^{13}.
\][/tex]
The coefficient is correct (24) and the exponent of [tex]$x$[/tex] matches ([tex]$x^6$[/tex]), but the [tex]$y$[/tex] exponent is [tex]$13$[/tex] instead of the required [tex]$15$[/tex].
--------------------------------------------------------------------
Option 2: Dimensions are
[tex]$$
6x^2y^3 \quad \text{and} \quad 4x^3y^5.
$$[/tex]
Multiply these dimensions:
[tex]\[
(6x^2y^3) \cdot (4x^3y^5) = 6 \cdot 4 \cdot x^{2+3} \cdot y^{3+5} = 24x^5y^8.
\][/tex]
The coefficient is 24, but the exponents are off: we get [tex]$x^5$[/tex] (instead of [tex]$x^6$[/tex]) and [tex]$y^8$[/tex] (instead of [tex]$y^{15}$[/tex]).
--------------------------------------------------------------------
Option 3: Dimensions are
[tex]$$
10x^6y^{15} \quad \text{and} \quad 14x^6y^{15}.
$$[/tex]
Their product is:
[tex]\[
(10x^6y^{15}) \cdot (14x^6y^{15}) = 10 \cdot 14 \cdot x^{6+6} \cdot y^{15+15} = 140x^{12}y^{30}.
\][/tex]
Here the coefficient is 140, and both exponents are much larger than required.
--------------------------------------------------------------------
Option 4: Dimensions are
[tex]$$
9x^4y^{11} \quad \text{and} \quad 12x^2y^4.
$$[/tex]
Multiply them:
[tex]\[
(9x^4y^{11}) \cdot (12x^2y^4) = 9 \cdot 12 \cdot x^{4+2} \cdot y^{11+4} = 108x^6y^{15}.
\][/tex]
While the exponents of [tex]$x$[/tex] and [tex]$y$[/tex] match the given expression exactly, the coefficient is 108 rather than 24.
--------------------------------------------------------------------
Reviewing the results:
- Option 1: [tex]$24x^6y^{13}$[/tex]
- Option 2: [tex]$24x^5y^8$[/tex]
- Option 3: [tex]$140x^{12}y^{30}$[/tex]
- Option 4: [tex]$108x^6y^{15}$[/tex]
None of the options exactly multiply to [tex]$24x^6y^{15}$[/tex]. However, Option 1 has the correct numerical coefficient (24) and the correct exponent for [tex]$x$[/tex] ([tex]$x^6$[/tex]). The discrepancy lies only in the [tex]$y$[/tex] exponent where [tex]$y^{13}$[/tex] is produced rather than the required [tex]$y^{15}$[/tex]. Given that Option 1 is the only one that has the correct coefficient and [tex]$x$[/tex] exponent, it is the most reasonable choice.
Thus, the dimensions of the rectangle are:
[tex]$$
\boxed{2x^5y^6 \text{ and } 12xy^7.}
$$[/tex]
