Final answer:
The width of the rectangle is the greatest common monomial factor of 70y⁸ and 30y⁶, which is 10y⁶. To find the length, divide the area by the width, resulting in a length of 7y² + 3. The rectangle's dimensions are width 10y⁶ and length 7y² + 3.
Step-by-step explanation:
To find the length and width of the rectangle with an area of 70y⁸ 30y⁶, we first need to determine the width by finding the greatest common monomial factor of the given terms. The greatest common factor (GCF) of the numerical coefficients (70 and 30) is 10, and the GCF of the variables y⁸ and y⁶ is y⁶ since it's the highest power of y that divides both terms. Therefore, the width of the rectangle is 10y⁶.
To find the length, we divide the area of the rectangle by the width. So the length is 70y⁸ / 10y⁶ = 7y² and 30y⁶ / 10y⁶ = 3. Combining these, the length of the rectangle is 7y² + 3.
The dimensions of the rectangle are therefore a width of 10y⁶ and a length of 7y² + 3.