Answer :
To find the area of the base of the cylinder in terms of x and π, we need to remember the formula for the volume of a cylinder:
\[ \text{Volume of a cylinder} = \pi \times \text{radius}^2 \times \text{height} \]
Given that the volume of the cylinder is represented by π(x³ + 32x² - 304x + 640) and the height of the cylinder is x + 40 feet, we can first express the volume formula using the given values:
\[ \pi \times \text{radius}^2 \times (x + 40) = \pi(x³ + 32x² - 304x + 640) \]
Now, we can equate the two expressions to find the radius of the cylinder:
\[ \text{radius}^2 = x³ + 32x² - 304x + 640 \div (x + 40) \]
\[ \text{radius}^2 = x² + 8x - 16 \]
Finally, the area of the base of the cylinder is given by the formula:
\[ \text{Area of base} = \pi \times \text{radius}^2 \]
Substitute the expression for the radius we found earlier into this formula to get the area of the base in terms of x and π:
\[ \text{Area of base} = \pi \times (x² + 8x - 16) \]
