31. VOLUME The volume of a cylinder is π(x³ + 32x² - 304x + 640). If the height of
the cylinder is x + 40 feet, find the area of its base in terms of x and T.



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: \[ \text{Area of base} = \pi \times (x² + 8x - 16) \]