Problem 39 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A vitamin capsule is constructed from a cylinder of length \(x\) centimeters and radius \(r\) centimeters, capped on either end by a hemisphere, as shown at left. Suppose that the length of the cylinder is equal to three times the diameter of the hemispherical caps. (a) Express the volume of the vitamin capsule as a function of \(x .\) Your strategy should be to begin by expressing the volume as a function of both \(x\) and \(r\). (b) Express the surface area of the vitamin capsule as a function of \(x\).

Step-by-Step Solution

Answer

The volume of the vitamin capsule as a function of \(x\) is \(V(x) = \pi x^3 /24\) cubic centimeters and the surface area of the vitamin capsule as a function of \(x\) is \(A(x) = \pi x^2 /2\) square centimeters.

Step 1: Express the Volume

Express the volume \(V\) of the vitamin capsule by adding up the volume of the cylindrical section and the two hemispherical ends. The volume of the cylinder is \(V_{cylinder} = \pi r^2 x\), and the volume of a hemisphere is \(V_{hemisphere} = 2/3 \pi r^3\). So the volume of the capsule is \(V = V_{cylinder} + 2 \cdot V_{hemisphere} = \pi r^2 x + 2 \cdot 2/3 \pi r^3 = \pi r^2 x +4/3 \pi r^3\). Express \(r\) in terms of \(x\) since it is given that \(x = 3 \cdot 2r = 6r\), therefore \(r = x/6\). Substitute \(r = x/6\) into the volume formula to get \(V(x) = \pi (x/6)^2 x +4/3 \pi (x/6)^3\).

Step 2: Simplify

Simplify the functions to find volume in terms of \(x\). \(V(x) = \pi (x/6)^2 x +4/3 \pi (x/6)^3 = \pi x^3/36 +4/3 \pi x^3 /216 = \pi x^3 /36 + \pi x^3 /162 = (\pi x^3) (1/36 + 1/162) = (\pi x^3) /24. So, the volume of the capsule is \(V(x) = \pi x^3 /24\) cubic centimeters.

Step 3: Express the Surface Area

Express the surface area \(A\) similarly as we did with the volume by adding the surface area of the cylinder and the two hemispheres. The surface area of the cylinder is \(A_{cylinder} = 2 \pi r x\) and the surface area of a hemisphere is \(A_{hemisphere} = 2 \pi r^2\). The surface area is consequently, \(A = A_{cylinder} + 2 \cdot A_{hemisphere} = 2 \pi r x + 2 \cdot 2 \pi r^2 = 2 \pi r x + 4 \pi r^2\). Use the fact that \(r = x/6\), the surface area in terms of \(x\) becomes \(A(x) = 2 \pi (x/6) x + 4 \pi (x/6)^2\).

Step 4: Simplify

Simplify the functions to find surface area in terms of \(x\). \(A(x) = 2 \pi (x/6) x + 4 \pi (x/6)^2 = 2 \pi x^2 /6 + 4 \pi x^2 /36 = \pi x^2 /3 + \pi x^2 /9 = (\pi x^2) (1/3 + 1/9) = (\pi x^2) /2. So, the surface area of the capsule is \(A(x) = \pi x^2 /2\) square centimeters.

Key Concepts

Solid Geometry

Solid geometry is the branch of mathematics that studies the properties and relationships of three-dimensional shapes. It is a crucial foundation for understanding the real world, as it describes the shapes and volumes of everyday objects, from simple ones like spheres and cylinders to complex structures like buildings and machinery. Calculating the volume and surface area of capsules involves two basic shapes from solid geometry: the cylinder and the hemisphere. Understanding these individual components is essential in determining the total volume and surface area of a combined shape, such as a vitamin capsule.

Students might find that visualizing these objects and breaking them down into their simpler parts can aid in comprehending how their dimensions relate to formulas for volume and surface area. Solid geometry also teaches us the significance of mathematical relationships between various shapes and how to modify equations when dimensions change, as seen in the process of substituting the radius in terms of length for the capsule exercise.

Cylinder Volume

The volume of a cylinder is found by multiplying the area of its circular base by its height. The base has a radius (\r), and the formula for the area A of a circle is \( A = \rho r^2 \) where \( \rho \) is the constant Pi (approximately 3.14159). The height (\r) of the cylinder is the distance between the two bases.

Therefore, the volume \( V \) of a cylinder with radius \( r \) and height \( x \) is calculated using the formula \( V_{cylinder} = \rho r^2 x \). For students, remembering this formula and understanding it as a product of base area and height is imperative for calculating the volume of cylindrical objects.

Hemisphere Volume

A hemisphere is half of a sphere, and the volume of a sphere is \( V_{sphere} = \rho r^3 \) where \( r \) is the radius. To find the volume of a hemisphere, you simply halve the volume of a sphere, resulting in the formula \( V_{hemisphere} = \rho (2/3)r^3 \).

By recognizing that a hemisphere is a portion of a sphere, students can see how shapes can be deconstructed into simpler parts to make volume calculation more manageable. This knowledge is particularly useful when combined shapes, like the vitamin capsule, include hemispherical components―in this case, constituting the ends of the capsule.

Surface Area Calculation

To calculate the surface area of a three-dimensional object, you add up the area of all the shapes that cover its surface. In the case of a cylinder, the surface area is composed of two circles (the bases) and the side surface, which, when unrolled, is a rectangle. The combined surface area of a hemisphere includes the curved outer surface of half a sphere.

The formula to calculate the surface area of a cylinder with radius \( r \) and height \( x \) is \( A_{cylinder} = 2 \rho rx \), representing the side surface, plus \( 2 \rho r^2 \) for the two bases. For a hemisphere, the formula is \( A_{hemisphere} = 2 \rho r^2 \) for just the curved surface. Since a vitamin capsule has two hemispherical ends, its total surface area combines these two components, leading to the equation provided in the textbook solution.