### Instruction: Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches?### Response: I need to find the original radius of the cylinder, which I will call r.
I also need to know the formula for the volume of a cylinder, which is V = pi * r^2 * h, where h is the height.
The problem says that increasing the radius or the height by six inches will result in the same volume, so I can write an equation to represent this: pi * (r + 6)^2 * 2 = pi * r^2 * (2 + 6).
I can simplify this equation by dividing both sides by pi and expanding the squares: (r + 6)^2 * 2 = r^2 * 8.
Further simplifying, I get 2r^2 + 24r + 72 = 8r^2.
Subtracting 2r^2 from both sides, I get 6r^2 - 24r - 72 = 0.
This is a quadratic equation, which I can solve by factoring: 6(r - 6)(r + 2) = 0.
The solutions are r = 6 and r = -2, but I can ignore the negative one since it does not make sense for a radius.
Therefore, the original radius is 6 inches.