Answer :
- Multiply each pair of given dimensions.
- Check if the result matches the given area $24x^6y^{15}$.
- Pair 1: $(2x^5y^8)(12xy^7) = 24x^6y^{15}$.
- The correct dimensions are $2x^5y^8$ and $12xy^7$. $\boxed{2 x^5 y^8 \text{ and } 12 x y^7}$
### Explanation
1. Understanding the Problem
We are given the area of a rectangle as $A = 24x^6y^{15}$ and four possible pairs of dimensions. We need to find the pair of dimensions that, when multiplied, gives us the area $A$.
2. Checking Each Pair of Dimensions
Let's check each pair of dimensions:
Pair 1: $(2x^5y^8)(12xy^7) = (2 \times 12)(x^5 Imes x)(y^8 Imes y^7) = 24x^{5+1}y^{8+7} = 24x^6y^{15}$
Pair 2: $(6x^2y^3)(4x^3y^5) = (6 \times 4)(x^2 \times x^3)(y^3 \times y^5) = 24x^{2+3}y^{3+5} = 24x^5y^8$
Pair 3: $(10x^6y^{15})(14x^6y^{15}) = (10 \times 14)(x^6 \times x^6)(y^{15} \times y^{15}) = 140x^{6+6}y^{15+15} = 140x^{12}y^{30}$
Pair 4: $(9x^4y^{11})(12x^2y^4) = (9 \times 12)(x^4 \times x^2)(y^{11} \times y^4) = 108x^{4+2}y^{11+4} = 108x^6y^{15}$
3. Identifying the Correct Dimensions
From the calculations above, we can see that only Pair 1, $(2x^5y^8)$ and $(12xy^7)$, results in the area $24x^6y^{15}$ when multiplied.
4. Final Answer
Therefore, the dimensions of the rectangle could be $2x^5y^8$ and $12xy^7$.
### Examples
Understanding how to calculate the area of a rectangle and manipulate algebraic expressions is useful in many real-world scenarios. For example, if you're designing a rectangular garden and know the total area you want it to cover, you can use this knowledge to determine the possible lengths and widths of the garden. Similarly, in construction or interior design, calculating areas and dimensions is crucial for estimating material quantities and ensuring that objects fit properly within a space. This problem demonstrates a fundamental concept in algebra and geometry that has practical applications in various fields.
- Check if the result matches the given area $24x^6y^{15}$.
- Pair 1: $(2x^5y^8)(12xy^7) = 24x^6y^{15}$.
- The correct dimensions are $2x^5y^8$ and $12xy^7$. $\boxed{2 x^5 y^8 \text{ and } 12 x y^7}$
### Explanation
1. Understanding the Problem
We are given the area of a rectangle as $A = 24x^6y^{15}$ and four possible pairs of dimensions. We need to find the pair of dimensions that, when multiplied, gives us the area $A$.
2. Checking Each Pair of Dimensions
Let's check each pair of dimensions:
Pair 1: $(2x^5y^8)(12xy^7) = (2 \times 12)(x^5 Imes x)(y^8 Imes y^7) = 24x^{5+1}y^{8+7} = 24x^6y^{15}$
Pair 2: $(6x^2y^3)(4x^3y^5) = (6 \times 4)(x^2 \times x^3)(y^3 \times y^5) = 24x^{2+3}y^{3+5} = 24x^5y^8$
Pair 3: $(10x^6y^{15})(14x^6y^{15}) = (10 \times 14)(x^6 \times x^6)(y^{15} \times y^{15}) = 140x^{6+6}y^{15+15} = 140x^{12}y^{30}$
Pair 4: $(9x^4y^{11})(12x^2y^4) = (9 \times 12)(x^4 \times x^2)(y^{11} \times y^4) = 108x^{4+2}y^{11+4} = 108x^6y^{15}$
3. Identifying the Correct Dimensions
From the calculations above, we can see that only Pair 1, $(2x^5y^8)$ and $(12xy^7)$, results in the area $24x^6y^{15}$ when multiplied.
4. Final Answer
Therefore, the dimensions of the rectangle could be $2x^5y^8$ and $12xy^7$.
### Examples
Understanding how to calculate the area of a rectangle and manipulate algebraic expressions is useful in many real-world scenarios. For example, if you're designing a rectangular garden and know the total area you want it to cover, you can use this knowledge to determine the possible lengths and widths of the garden. Similarly, in construction or interior design, calculating areas and dimensions is crucial for estimating material quantities and ensuring that objects fit properly within a space. This problem demonstrates a fundamental concept in algebra and geometry that has practical applications in various fields.
