Problem 80 - Calculus Early Transcendentals

Question

Bob and Bruce bake bagels (shaped like tori). They both make bagels that have an inner radius of 0.5 in and an outer radius of 2.5 in. Bob plans to increase the volume of his bagels by decreasing the inner radius by \(20 \%\) (leaving the outer radius unchanged). Bruce plans to increase the volume of his bagels by increasing the outer radius by \(20 \%\) (leaving the inner radius unchanged). Whose new bagels will have the greater volume? Does this result depend on the size of the original bagels? Explain.

Step-by-Step Solution

Answer

Answer: Bruce's new bagels will have the greater volume after increasing the outer radius by 20%.

Step 1: Original Bagel Dimensions and Volume Calculation

Given bagel dimensions: Inner radius = 0.5 in Outer radius = 2.5 in The distance from the center of the hole to the center of the tube (R) is \((\frac{outer \thinspace radius + inner \thinspace radius}{2})\), After calculating, we have R = 1.5 in. The radius of the tube (r) is \((\frac{outer \thinspace radius - inner \thinspace radius}{2})\). After calculating, we have r = 1 in. We have the formula for the volume of a torus, V = \(2π^2Rr^2\). Plugging in our values: Original Bagel Volume = \(2π^2(1.5)(1)^2\) = \(9π\) cubic inches.

Step 2: New Bagel Dimensions for Bob and Bruce

Now, we'll calculate the revised dimensions for Bob and Bruce's bagels: Bob's Bagel: - New inner radius = \(0.5 - 0.5×0.2\) = 0.4 in (decrease by 20%) - Outer radius unchanged = 2.5 in For Bob's bagel, R = \((\frac{2.5+0.4}{2})\) = 1.45 in and r = \((\frac{2.5-0.4}{2})\) = 1.05 in. Bruce's Bagel: - Inner radius unchanged = 0.5 in - New outer radius = \(2.5 + 2.5×0.2\) = 3 in (increase by 20%) For Bruce's bagel, R = \((\frac{3+0.5}{2})\) = 1.75 in and r = \((\frac{3-0.5}{2})\) = 1.25 in.

Step 3: New Bagel Volume Calculation and Comparison

Now, we'll calculate the new volumes for Bob and Bruce's bagels: Bob's New Bagel Volume = \(2π^2(1.45)(1.05)^2\) = \(10.18275π\) cubic inches. Bruce's New Bagel Volume = \(2π^2(1.75)(1.25)^2\) = \(15.359375π\) cubic inches. Comparing the two volumes, we can see that Bruce's new bagels have a greater volume (15.359375π cubic inches) than Bob's new bagels (10.18275π cubic inches).

Step 4: Conclusion

Bruce's new bagels have the greater volume after increasing the outer radius by 20%. This result does not depend on the size of the original bagels because increasing the outer radius contributes more to increasing the volume of the bagels than decreasing the inner radius. Bob's inner radius reduction remaining outer radius intact so the impact of increasing volume is less than Bruce's increasing of outer radius remaining inner radius intact which stretched both radii R and r.

Key Concepts

Volume of a Torus

Understanding the volume of a torus, which is a shape resembling a doughnut or, in our delicious analogy, a bagel, is fundamental in various fields, from engineering to gastronomy. The formula for the volume of a torus is given by \( V = 2\pi^2Rr^2 \), where \( R \) is the distance from the center of the hole to the center of the tube, and \( r \) is the radius of the tube itself.

Imagine slicing the torus into circular cross-sections. Each circle's area is \( \pi r^2 \), and as we rotate this area around the central axis of the torus for one full rotation, the resulting shape is the torus. The factor of \( 2\pi \) in the volume formula accounts for this rotation and the even distribution of the tube's cross-sectional area around the axis.

For the problem at hand, Bob and Bruce's original bagels had volumes of \( 9\pi \) cubic inches. The formula showcases how the radii contribute to volume, emphasizing that even small changes to \( R \) or \( r \) will significantly affect the bagel's volume, a topic we'll explore further in the context of radius manipulation.

Radius Manipulation

Radius manipulation involves adjusting the radius (or radii) of an object to alter its dimensions and, subsequently, various properties such as volume. In the case of Bob and Bruce's bagels, radius manipulation was executed differently by each baker to increase the volume.

For Bob, decreasing the inner radius by \(20\%\) while keeping the outer radius unchanged altered both \( R \) and \( r \), but in a subtle way. Bob's reduction of the inner radius, resulting in an \( R \) of 1.45 inches and an \( r \) of 1.05 inches, shows a small but significant increase in volume.

On the other hand, Bruce chose to increase the outer radius, which had a more dramatic effect on both \( R \) and \( r \). The larger increase in both radii for Bruce's bagel, to an \( R \) of 1.75 inches and an \( r \) of 1.25 inches, produced a more substantial volume increase.

This illustrates an important concept in geometry: a slight alteration in dimensions can have a major impact on volume, particularly as the volume formula involves squaring one of the radii (\( r^2 \)). Therefore, students should be keenly aware that a delicate balance of radius manipulation can profoundly influence a shape's capacity.

Increasing Volume Strategies

There are several strategies for increasing the volume of a three-dimensional object, and the problem of Bob and Bruce's bagels perfectly illustrates two different approaches within the context of the torus.

The strategy that Bruce implemented is a prime example of how increasing the outer radius (while keeping the inner radius constant) yields a larger effect on volume. This is because the increase affects both \( R \) and \( r \), and since \( r \) is squared in the volume formula, the impact of altering it is amplified.

Bruce's strategy was clearly more effective, as his volume increased more substantially compared to Bob's. This teaches us an important lesson in volume optimization: focusing on outer geometry adjustments can often lead to a greater impact on volume than inner geometry adjustments.

These strategies also highlight a broader implication for students and practitioners alike: understanding and applying mathematical concepts allows effective problem-solving in real-world scenarios. Whether it's optimizing the design of a product or getting the most out of a baking endeavor, the principles of calculus and geometry hold the key to success.