Answer :
To solve this problem, we need to determine which of the given pairs of expressions for dimensions, when multiplied together, give the area of the rectangle as [tex]\(24 x^6 y^{15}\)[/tex].
The area of a rectangle is calculated by multiplying its length and width. So, we need to multiply each of the given pairs of dimensions and see if any resulting expression equals [tex]\(24 x^6 y^{15}\)[/tex].
### Let's evaluate each option:
1. Option 1: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 12 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
The product is [tex]\(24 x^6 y^{15}\)[/tex]. This matches the given area expression.
2. Option 2: [tex]\(6 x^2 y^3\)[/tex] and [tex]\(4 x^3 y^5\)[/tex]
- Multiply the coefficients: [tex]\(6 \times 4 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
The product is [tex]\(24 x^5 y^8\)[/tex]. This does not match the given area expression.
3. Option 3: [tex]\(10 x^6 y^{15}\)[/tex] and [tex]\(14 x^6 y^{15}\)[/tex]
- Multiply the coefficients: [tex]\(10 \times 14 = 140\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
The product is [tex]\(140 x^{12} y^{30}\)[/tex]. This does not match the given area expression.
4. Option 4: [tex]\(9 x^4 y^{11}\)[/tex] and [tex]\(12 x^2 y^4\)[/tex]
- Multiply the coefficients: [tex]\(9 \times 12 = 108\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
The product is [tex]\(108 x^6 y^{15}\)[/tex]. This does not match the given area expression.
### Conclusion:
The dimensions that result in the area expression [tex]\(24 x^6 y^{15}\)[/tex] are:
Option 1: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
The area of a rectangle is calculated by multiplying its length and width. So, we need to multiply each of the given pairs of dimensions and see if any resulting expression equals [tex]\(24 x^6 y^{15}\)[/tex].
### Let's evaluate each option:
1. Option 1: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 12 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
The product is [tex]\(24 x^6 y^{15}\)[/tex]. This matches the given area expression.
2. Option 2: [tex]\(6 x^2 y^3\)[/tex] and [tex]\(4 x^3 y^5\)[/tex]
- Multiply the coefficients: [tex]\(6 \times 4 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
The product is [tex]\(24 x^5 y^8\)[/tex]. This does not match the given area expression.
3. Option 3: [tex]\(10 x^6 y^{15}\)[/tex] and [tex]\(14 x^6 y^{15}\)[/tex]
- Multiply the coefficients: [tex]\(10 \times 14 = 140\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
The product is [tex]\(140 x^{12} y^{30}\)[/tex]. This does not match the given area expression.
4. Option 4: [tex]\(9 x^4 y^{11}\)[/tex] and [tex]\(12 x^2 y^4\)[/tex]
- Multiply the coefficients: [tex]\(9 \times 12 = 108\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
The product is [tex]\(108 x^6 y^{15}\)[/tex]. This does not match the given area expression.
### Conclusion:
The dimensions that result in the area expression [tex]\(24 x^6 y^{15}\)[/tex] are:
Option 1: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
