Problem 55 - Thomas Calculus

Question

Designing a wok You are designing a wok frying pan that will be shaped like a spherical bowl with handles. A bit of experimentation at home persuades you that you can get one that holds about 3 \(\mathrm{L}\) if you make it 9 \(\mathrm{cm}\) deep and give the sphere a radius of \(16 \mathrm{cm} .\) To be sure, you picture the wok as a solid of revolution, as shown here, and calculate its volume with an integral. To the nearest cubic centimeter, what volume do you really get? \(\left(1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right)\)

Step-by-Step Solution

Answer

The wok's volume is approximately 3312 cubic centimeters.

Step 1: Identify the Volume of a Sphere Segment

A wok can be thought of as part of a sphere. First, identify that the wok will be created from the volume of a spherical cap. The sphere's radius is given as 16 cm, and the wok's depth is 9 cm.

Step 2: Apply the Formula for the Volume of a Spherical Cap

The volume of a spherical cap can be calculated with the formula \( V = \frac{1}{3}\pi h^2 (3R - h) \), where \( h \) is the depth of the cap (9 cm) and \( R \) is the sphere's radius (16 cm).

Step 3: Substitute the Values into the Formula

Substitute \( h = 9 \text{ cm} \) and \( R = 16 \text{ cm} \) into the formula: \[ V = \frac{1}{3}\pi (9)^2 (3(16) - 9) \].

Step 4: Calculate the Expression for Volume

Simplify the expression: 1. Calculate \( 9^2 = 81 \).2. Calculate \( 3 \times 16 = 48 \).3. Then, \( 48 - 9 = 39 \).4. Compute the product: \( 81 \times 39 = 3159 \).5. Multiply this by \( \pi \) and divide by 3: \( V = \frac{1}{3} \times \pi \times 3159 \approx 3312.41 \text{ cm}^3 \).

Step 5: Round the Final Volume to the Nearest Cubic Centimeter

Round 3312.41 to the nearest whole number, which gives approximately 3312 cubic centimeters.

Key Concepts

Solid of Revolution

A solid of revolution is a 3D object created by rotating a 2D shape around an axis. This idea is useful in calculating volumes of objects like woks, bowls, and other curved items. Imagine taking the profile of a semicircle and spinning it around its generally vertical axis. The result is a spherical shape, much like the curved part of a wok.
A wok, as described in the exercise, forms part of a sphere called a spherical cap.
When designing a wok as a solid of revolution, you consider how the curved surface wraps around. Thus, you can use calculus techniques to find its volume by integrating the area of its cross-sections. This rotational method offers a practical way to handle complex shapes figuring out spaciousness without direct measurement.
Understanding solids of revolution is key to grasping how integral calculus converts 2D dimensions to 3D volumes, bridging abstract concepts with real-world objects.

Integral Calculus

Integral calculus is a mathematical tool used to calculate areas under curves, among other things. In the case of finding volumes, it's especially helpful with solids of revolution. By integrating along an axis, you can accumulate "slices" of the shape to find the total volume.
Imagine slicing the spherical cap into numerous thin discs. By adding these discs' volumes together using integration, you determine how much space the cap occupies.
This process involves first determining the function representing the shape's boundary and then calculating its integral over the rotation's limits. For the wok example, you calculated the integral of a spherical cap using a specific formula that applies precisely to this scenario.
Integral calculus, therefore, allows us to compute complex volumes systematically, transforming abstract mathematical functions into tangible dimensions.

Formula for Volume

The formula for calculating the volume of a spherical cap is a crucial tool in solving the exercise. The spherical cap's formula is given by:
\[ V = \frac{1}{3}\pi h^2 (3R - h) \]
where \( h \) is the height or depth (9 cm for the wok), and \( R \) is the sphere's radius (16 cm). This formula calculates the volume of the region defined between a plane cutting through the sphere and its surface.
Each parameter here represents a physical dimension of the spherical shape:

• \( h \) relates to how deep the cut is into the sphere, affecting the cap's height.
• \( R \) informs how large the initial sphere is, impacting the cap's size.

When you integrate these dimensions with the formula, it efficiently gives the volume of the spherical cap.
Following these calculations simplifies determining the wok's functionality and capacity. This formula is beneficial because it provides precise results without needing cumbersome geometric measurements directly.