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Answer :
* The problem requires finding the dimensions of a rectangle given its area $A = 24x^6y^{15}$.
* We test each option by multiplying the given dimensions.
* Option 1: $(2x^5y^8)(12xy^7) = 24x^6y^{15}$, which matches the given area.
* Therefore, the dimensions could be $2x^5y^8$ and $12xy^7$. $\boxed{2 x^5 y^8 \text{ and } 12 x y^7}$
### Explanation
1. Problem Analysis
We are given the area of a rectangle as $A = 24x^6y^{15}$ and asked to find which of the given pairs of dimensions could be the length and width of the rectangle. To do this, we will multiply each pair of dimensions and see if the result matches the given area.
2. Checking Option 1
Let's examine the first option: $2x^5y^8$ and $12xy^7$. Multiplying these gives $(2x^5y^8)(12xy^7) = 2 \cdot 12 \cdot x^5
\cdot x
\cdot y^8
\cdot y^7 = 24x^{5+1}y^{8+7} = 24x^6y^{15}$. This matches the given area.
3. Checking Option 2
Let's examine the second option: $6x^2y^3$ and $4x^3y^5$. Multiplying these gives $(6x^2y^3)(4x^3y^5) = 6
\cdot 4
\cdot x^2
\cdot x^3
\cdot y^3
\cdot y^5 = 24x^{2+3}y^{3+5} = 24x^5y^8$. This does not match the given area.
4. Checking Option 3
Let's examine the third option: $10x^6y^{15}$ and $14x^6y^{15}$. Multiplying these gives $(10x^6y^{15})(14x^6y^{15}) = 10
\cdot 14
\cdot x^6
\cdot x^6
\cdot y^{15}
\cdot y^{15} = 140x^{6+6}y^{15+15} = 140x^{12}y^{30}$. This does not match the given area.
5. Checking Option 4
Let's examine the fourth option: $9x^4y^{11}$ and $12x^2y^4$. Multiplying these gives $(9x^4y^{11})(12x^2y^4) = 9
\cdot 12
\cdot x^4
\cdot x^2
\cdot y^{11}
\cdot y^4 = 108x^{4+2}y^{11+4} = 108x^6y^{15}$. This does not match the given area.
6. Final Answer
Only the first option, $2x^5y^8$ and $12xy^7$, results in the area $24x^6y^{15}$. Therefore, these could be the dimensions of the rectangle.
### Examples
Understanding how to calculate the area of a rectangle is useful in many real-world scenarios. For example, if you're planning to install new flooring in a rectangular room, you need to calculate the area to determine how much material to purchase. Similarly, if you're designing a rectangular garden, knowing the area helps you estimate the amount of soil and plants you'll need. This concept is also crucial in fields like architecture and construction, where precise area calculations are essential for planning and resource management.
* We test each option by multiplying the given dimensions.
* Option 1: $(2x^5y^8)(12xy^7) = 24x^6y^{15}$, which matches the given area.
* Therefore, the dimensions could be $2x^5y^8$ and $12xy^7$. $\boxed{2 x^5 y^8 \text{ and } 12 x y^7}$
### Explanation
1. Problem Analysis
We are given the area of a rectangle as $A = 24x^6y^{15}$ and asked to find which of the given pairs of dimensions could be the length and width of the rectangle. To do this, we will multiply each pair of dimensions and see if the result matches the given area.
2. Checking Option 1
Let's examine the first option: $2x^5y^8$ and $12xy^7$. Multiplying these gives $(2x^5y^8)(12xy^7) = 2 \cdot 12 \cdot x^5
\cdot x
\cdot y^8
\cdot y^7 = 24x^{5+1}y^{8+7} = 24x^6y^{15}$. This matches the given area.
3. Checking Option 2
Let's examine the second option: $6x^2y^3$ and $4x^3y^5$. Multiplying these gives $(6x^2y^3)(4x^3y^5) = 6
\cdot 4
\cdot x^2
\cdot x^3
\cdot y^3
\cdot y^5 = 24x^{2+3}y^{3+5} = 24x^5y^8$. This does not match the given area.
4. Checking Option 3
Let's examine the third option: $10x^6y^{15}$ and $14x^6y^{15}$. Multiplying these gives $(10x^6y^{15})(14x^6y^{15}) = 10
\cdot 14
\cdot x^6
\cdot x^6
\cdot y^{15}
\cdot y^{15} = 140x^{6+6}y^{15+15} = 140x^{12}y^{30}$. This does not match the given area.
5. Checking Option 4
Let's examine the fourth option: $9x^4y^{11}$ and $12x^2y^4$. Multiplying these gives $(9x^4y^{11})(12x^2y^4) = 9
\cdot 12
\cdot x^4
\cdot x^2
\cdot y^{11}
\cdot y^4 = 108x^{4+2}y^{11+4} = 108x^6y^{15}$. This does not match the given area.
6. Final Answer
Only the first option, $2x^5y^8$ and $12xy^7$, results in the area $24x^6y^{15}$. Therefore, these could be the dimensions of the rectangle.
### Examples
Understanding how to calculate the area of a rectangle is useful in many real-world scenarios. For example, if you're planning to install new flooring in a rectangular room, you need to calculate the area to determine how much material to purchase. Similarly, if you're designing a rectangular garden, knowing the area helps you estimate the amount of soil and plants you'll need. This concept is also crucial in fields like architecture and construction, where precise area calculations are essential for planning and resource management.
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