Answer :
To find the dimensions of the rectangle whose area is given by [tex]\(24x^6y^{15}\)[/tex], we need to find which pair of dimension expressions, when multiplied, results in the same area expression. Let's go through each option one by one:
Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
1. Multiply the numerical coefficients: [tex]\(2 \times 12 = 24\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex].
So, the resulting area expression is [tex]\(24x^6y^{15}\)[/tex], which matches the given area.
Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
1. Multiply the numerical coefficients: [tex]\(6 \times 4 = 24\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex].
The resulting area is [tex]\(24x^5y^8\)[/tex], which does not match the given area.
Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
1. Multiply the numerical coefficients: [tex]\(10 \times 14 = 140\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex].
The resulting area is [tex]\(140x^{12}y^{30}\)[/tex], which does not match the given area.
Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
1. Multiply the numerical coefficients: [tex]\(9 \times 12 = 108\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex].
The resulting area is [tex]\(108x^6y^{15}\)[/tex], which does not match the given area.
After comparing all the options, Option 1 is the correct choice since the product equals the given area, [tex]\(24x^6y^{15}\)[/tex]. Therefore, the dimensions of the rectangle are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
1. Multiply the numerical coefficients: [tex]\(2 \times 12 = 24\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex].
So, the resulting area expression is [tex]\(24x^6y^{15}\)[/tex], which matches the given area.
Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
1. Multiply the numerical coefficients: [tex]\(6 \times 4 = 24\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex].
The resulting area is [tex]\(24x^5y^8\)[/tex], which does not match the given area.
Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
1. Multiply the numerical coefficients: [tex]\(10 \times 14 = 140\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex].
The resulting area is [tex]\(140x^{12}y^{30}\)[/tex], which does not match the given area.
Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
1. Multiply the numerical coefficients: [tex]\(9 \times 12 = 108\)[/tex].
2. Multiply the [tex]\(x\)[/tex] terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex].
3. Multiply the [tex]\(y\)[/tex] terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex].
The resulting area is [tex]\(108x^6y^{15}\)[/tex], which does not match the given area.
After comparing all the options, Option 1 is the correct choice since the product equals the given area, [tex]\(24x^6y^{15}\)[/tex]. Therefore, the dimensions of the rectangle are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
