Problem 59 - Calculus Early Transcendentals

Question

The volume of a solid torus (a bagel or doughnut) is given by \(V=\left(\pi^{2} / 4\right)(R+r)(R-r)^{2},\) where \(r\) and \(R\) are the inner and outer radii and \(R>r\) (see figure). a. If \(R\) and \(r\) increase at the same rate, does the volume of the torus increase, decrease, or remain constant? b. If \(R\) and \(r\) decrease at the same rate, does the volume of the torus increase, decrease, or remain constant?

Step-by-Step Solution

Answer

Answer: When both radii increase at the same rate, the volume of the torus increases, and when both radii decrease at the same rate, the volume of the torus decreases.

Step 1: Observe the volume formula

The formula for the volume of the torus is given by \(V=\left(\frac{\pi^{2}}{4}\right)(R+r)(R-r)^{2},\) where \(R\) and \(r\) are the outer and inner radii, respectively, and \(R>r\).

Step 2: Write down the conditions on radii

In order to investigate the changes in volume when different conditions are applied to the radii, let's represent the changes by two variables: \(R'=R+c\) and \(r'=r+c\), where \(c>0\) represents the rate of change. Note that we use the same value \(c\) in both cases, as the problem states that the radii change at the same rate.

Step 3: Determine the impact of increasing the radii on the volume

If both radii increase at the same rate, we have the new volume formula: \(V'=\left(\frac{\pi^{2}}{4}\right)((R+c)+(r+c))((R+c)-(r+c))^{2}\). Simplify the expression: \(V'=\left(\frac{\pi^{2}}{4}\right)(R+r+2c)(R-r)^{2}\). Comparing to the original volume formula, it can be observed that the value of \(2c\) has been added to the original expression. Therefore, if \(c>0\), the volume of the torus will increase when both radii increase at the same rate.

Step 4: Determine the impact of decreasing the radii on the volume

If both radii decrease at the same rate, we have the new volume formula: \(V'=\left(\frac{\pi^{2}}{4}\right)((R-c)+(r-c))((R-c)-(r-c))^{2}\). Simplify the expression: \(V'=\left(\frac{\pi^{2}}{4}\right)(R+r-2c)(R-r)^{2}\). Comparing to the original volume formula, it can be observed that the value of \(2c\) has been subtracted from the original expression. Therefore, if \(c>0\), the volume of the torus will decrease when both radii decrease at the same rate.

Step 5: Answers:

a. When both radii \(R\) and \(r\) increase at the same rate, the volume of the torus increases. b. When both radii \(R\) and \(r\) decrease at the same rate, the volume of the torus decreases.

Key Concepts

Volume of a Solid

The volume of a solid object tells us how much space it occupies. For a torus, which resembles a doughnut or bagel, the volume is calculated using a special formula. This formula takes into account the size of the hole (inner radius \( r \)) and the size of the entire torus (outer radius \( R \)). The formula given for the volume of a torus is: \[ V = \left(\frac{\pi^{2}}{4}\right)(R+r)(R-r)^{2} \]Here:

• \( \pi \) is a constant approximately equal to 3.14159,
• \( R \) is the outer radius,
• \( r \) is the inner radius.

The formula shows that the volume of the torus depends directly on both \( R \) and \( r \). If you increase these values, the volume will change. Understanding how these changes affect the volume is crucial, and we'll explore that idea through the rate of change concept.

Rate of Change

The rate of change in this context refers to how quickly or slowly a value, like a radius, is increasing or decreasing. When both the outer radius \( R \) and the inner radius \( r \) of a torus change at the same rate, it's important to determine how this affects the volume.Let's suppose both radii change by a small amount, represented by \( c \). If \( c > 0 \), both radii are increasing; if \( c < 0 \), they're decreasing. The changes can be incorporated into the volume formula. For increases, we have: \[ V' = \left(\frac{\pi^{2}}{4}\right)(R+r+2c)(R-r)^{2} \]For decreases, it becomes: \[ V' = \left(\frac{\pi^{2}}{4}\right)(R+r-2c)(R-r)^{2} \]These adjustments show:

• If both radii increase gradually, the added term \(+2c\) means the volume grows.
• If both radii decrease, the subtracted term \(-2c\) tells us the volume shrinks.

This highlights how linked changes in geometry impact the torus's overall size.

Geometry of a Torus

Understanding the geometry of a torus involves visualizing its unique shape. A torus is essentially a three-dimensional surface formed by revolving a circle in space around an axis coplanar with the circle. This geometric shape can be similar to a circular lifebuoy or a doughnut. Two critical measurements define a torus's structure: the outer and inner radii.

• The **outer radius** \( R \) measures from the center of the torus to the center of the tube.
• The **inner radius** \( r \) measures from the center of the tube to the outer surface.

The space within this curved surface can be quantified by calculating the volume, which combines these two radius measurements uniquely, as seen in the volume formula provided earlier.The torus's geometry is precisely why the inner and outer radii effectively balance and affect the torus's volume. These measurements dictate how large or complex the curved surface of the torus can become, playing a crucial role in both physical models like hoops and theoretical mathematics applications.