Which variable(s) in the formula ` V=pi r^2 h` would you need to set as a constant to generate a linear and quadratic equation?

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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`V=pi r^2h` is the formula for the volume of a cylinder with base radius `r` and height `h.` Since `pi` is constant, the volume depends entirely on `r` and `h`, and to emphasize this we can write

`V(r,h)=pi r^2h.`

When `r` is held constant, `pi r^2` is now constant as well, so we can call it `c` for convenience. Then the formula for volume only depends on `h` and is of the form `V(h)=pi r^2h=ch,` and this describes a linear function of `h.`

When `h` is held constant, `pi h` is constant and we can call it `k.` Now the volume depends only on `r` , and we can write `V(r)=pi r^2h=kr^2,` which is a quadratic function of `r.`

Thus, setting `r` as a constant results in a linear equation, and setting `h` as a constant results in a quadratic equation. Another way of saying the same thing is that volume varies linearly with `h` and quadratically with `r.`

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