Answer:The rectangle does indeed exist with the given dimensions when (y = 2).
Step-by-step explanation:
Certainly! Let’s find the expressions for the length and width of the rectangle.
Given that the area of the rectangle is 14y^3 - 63y, we want to express the dimensions in terms of the greatest common factor (GCF).
1. First, let’s factor the expression for the area: [ \text{Area} = 14y^3 - 63y ] To find the GCF, we can factor out the common factor from both terms. The GCF of 14 and 63 is 7. Let’s factor it out: [ \text{Area} = 7y(2y^2 - 9) ]
2. Now we have the expression for the area in factored form. We know that one dimension (either length or width) is the GCF, which is (7y).
3. To find the other dimension, we divide the factored area expression by the GCF: [ \text{Width} = \frac{{\text{Area}}}{{\text{Length}}} = \frac{{7y(2y^2 - 9)}}{{7y}} ] Cancel out the common factor of (7y): [ \text{Width} = 2y^2 - 9 ]
Therefore, the expressions for the length and width of the rectangle are:
●Length: (7y)
●Width: (2y^2 - 9)
If we set (y = 2), we can check if the rectangle exists with these dimensions: [ \text{Area} = 14(2)^3 - 63(2) = 52 ]
The rectangle does indeed exist with the given dimensions when (y = 2).
