Problem 3 - Thomas Calculus

Question

Volume The radius \(r\) and height \(h\) of a right circular cylinder are related to the cylinder's volume \(V\) by the formula \(V=\pi r^{2} h .\) a. How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? b. How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? c. How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

Step-by-Step Solution

Answer

a. \(dV/dt = \pi r^2 dh/dt\); b. \(dV/dt = 2\pi rh dr/dt\); c. \(dV/dt = 2\pi r h dr/dt + \pi r^2 dh/dt\)."

Step 1: Understand the Given Information

We know the volume of a right circular cylinder is given by the formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. We need to find the relationship between the rate of change of volume \(dV/dt\) and the rates of change of radius \(dr/dt\) and height \(dh/dt\) under different conditions.

Step 2a: Derive Expression for \(dV/dt\) when \(r\) is constant

When the radius \(r\) is constant, \(dr/dt = 0\). Differentiate the volume formula with respect to time \(t\): \[ \frac{dV}{dt} = \frac{d}{dt} (\pi r^2 h) = \pi r^2 \frac{dh}{dt} \]Thus, \(dV/dt = \pi r^2 dh/dt\).

Step 2b: Derive Expression for \(dV/dt\) when \(h\) is constant

When the height \(h\) is constant, \(dh/dt = 0\). Differentiate the formula:\[ \frac{dV}{dt} = \frac{d}{dt} (\pi r^2 h) = 2\pi rh \frac{dr}{dt} \]Thus, \(dV/dt = 2\pi rh dr/dt\).

Step 2c: Derive Expression for \(dV/dt\) when neither \(r\) nor \(h\) is constant

If neither \(r\) nor \(h\) is constant, use the product rule for differentiation:\[ \frac{dV}{dt} = \frac{d}{dt} (\pi r^2 h) = \pi (2r \frac{dr}{dt} h + r^2 \frac{dh}{dt}) \]Thus, \(dV/dt = 2\pi r h dr/dt + \pi r^2 dh/dt\).

Key Concepts

Volume of Cylinder

The volume of a right circular cylinder is a fundamental concept in geometry and calculus. To find the volume, we use the formula:

• \[ V = \pi r^{2} h \]

Here, \(V\) is the volume, \(r\) is the radius of the base of the cylinder, and \(h\) is the height of the cylinder.
This formula essentially multiplies the base area (a circle with radius \(r\)) by the height \(h\). Hence, it calculates how much space the cylinder occupies.

• When the radius or height changes, it affects the volume, making this formula essential in related rates problems.

Remember, the area of a circle (the base of the cylinder) is calculated by \(\pi r^2\). Therefore, understanding this formula helps grasp how the size of a cylinder might change over time if its dimensions vary.

Differentiation

Differentiation is a core idea in calculus that lets us understand how functions change. When we talk about the rate of change, we mean differentiation. It helps us find how one quantity changes concerning another.
In this context, when we differentiate the volume \(V\) of a cylinder with respect to time \(t\), denoted as \(dV/dt\), we understand how the volume changes over time as either \(r\), \(h\), or both change.

• When differentiating with respect to time, we consider how each variable (either \(r\) or \(h\)) changes over time.

• For example, if the height of the cylinder is changing, \(h\), and \(r\) is constant, the differentiation relates directly, simplifying to \( \pi r^2 (dh/dt) \).
• If the radius is changing when the height is constant, differentiation involves \(2\pi rh \frac{dr}{dt}\).

Using differentiation, we calculate these changes and translate them into our understanding of the physical world, assisting in solving practical problems related to motion or growth.

Product Rule

The product rule is a vital component of differentiation, especially when dealing with the product of two changing quantities. In our example of the cylinder volume, the product rule is applied to the expression \(\pi r^2 h\) when neither \(r\) nor \(h\) is constant.
This rule states: if you have two functions \(u(t)\) and \(v(t)\), then their derivative \((uv)'\) is given by:

• \[ (uv)' = u'v + uv' \]

Applying this to our problem, \(r^2\) and \(h\) both depend on time, so we use the product rule:

• \[ \frac{dV}{dt} = 2\pi rh \frac{dr}{dt} + \pi r^2 \frac{dh}{dt} \]

• This expression shows how the volume's rate of change (\(dV/dt\)) depends on both \(dr/dt\) and \(dh/dt\).

Therefore, understanding and using the product rule in differentiation allows us to manage changes in composite functions, which is crucial in analyzing how different factors combine to influence a system's behavior over time.