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Chapter 13: Problem 6

The volume of a right circular cylinder with radius \(r\) and height \(h\) is \(V=\pi r^{2} h .\) Is the volume an increasing or decreasing function of the radius at a fixed height (assume \(r>0\) and \(h>0\) )?

Short Answer

Expert verified
Answer: The volume of a right circular cylinder is an increasing function of the radius when the height is fixed.

Step by step solution

01

Write down the formula for the volume of a right circular cylinder

The formula for the volume of a right circular cylinder is \(V=\pi r^{2}h\), where \(r\) is the radius and \(h\) is the height.
02

Differentiate the volume formula with respect to the radius

To find the derivative of the volume with respect to the radius, we will use the power rule. So, we get: \(\frac{dV}{dr} = \frac{d(\pi r^2 h)}{dr} = \pi h \frac{d(r^2)}{dr}\). Now, differentiate \(r^2\) with respect to \(r\): \(\frac{d(r^2)}{dr} = 2r\). So, the derivative of the volume with respect to the radius is: \(\frac{dV}{dr} = \pi h(2r) = 2\pi rh\).
03

Determine if the derivative is positive or negative

We need to look at the sign of the derivative \(\frac{dV}{dr} = 2\pi rh\) to know if the function is increasing or decreasing. Since \(r>0\) and \(h>0\) are both given, the expression \(2\pi rh\) is positive.
04

State the conclusion

Since the derivative \(\frac{dV}{dr}>0\), the volume of a right circular cylinder is an increasing function of the radius when the height is fixed.

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Most popular questions from this chapter

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