Problem 30 - Calculus Early Transcendentals

Question

The volume of a cone of radius \(r\) and height \(h\) is one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius \(r\) and height \(h\) equal one-third the surface area of a cylinder with the same radius and height? If not, find the correct relationship. Exclude the bases of the cone and cylinder.

Step-by-Step Solution

Answer

Answer: The volume of a cone is one-third the volume of a cylinder with the same height and radius. However, there is no constant relationship between their surface areas when disregarding the bases. The surface area relationship between a cone and a cylinder with the same height and radius is given by the formula: \( \frac{A_{cone}}{A_{cylinder}} = \frac{\sqrt{r^2 + h^2}}{2h} \).

Step 1: Recall Volume and Surface Area Formulas

The formulas for the volume and surface area of a cone and a cylinder are as follows: 1. Cone: - Volume: \(V=\frac{1}{3}\pi r^2 h\) - Surface Area: \(A=\pi r l + \pi r^2\) (including base) or \(A=\pi r l\) (excluding base), where \(l\) is the slant height 2. Cylinder: - Volume: \(V=\pi r^2 h\) - Surface Area: \(A=2\pi r h + 2\pi r^2\) (including base) or \(A=2\pi r h\) (excluding base)

Step 2: Prove Volume Relationship

Let's prove that the volume of a cone is one-third the volume of a cylinder with the same radius and height: - Cone Volume: \(V_{cone}=\frac{1}{3}\pi r^2 h\) - Cylinder Volume: \(V_{cylinder}=\pi r^2 h\) Now, divide the cone volume by the cylinder volume: \(\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h} = \frac{1}{3}\) Since the fraction is equal to \(\frac{1}{3}\), the volume of a cone is indeed one-third the volume of a cylinder with the same height and radius.

Step 3: Determine Surface Area Relationship

Now let's determine the relationship between the surface areas of a cone and a cylinder with the same radius and height, excluding the bases: 1. Cone Surface Area (excluding base): \(A_{cone}=\pi r l\) 2. Cylinder Surface Area (excluding base): \(A_{cylinder}=2\pi r h\) To find the slant height (\(l\)) of the cone, we can use the Pythagorean theorem in a right triangle composed by the radius, the height, and the slant height, \(l^2 = r^2 + h^2\), so \(l = \sqrt{r^2 + h^2}\). Now, substitute the slant height in the cone surface area formula: \(A_{cone}=\pi r (\sqrt{r^2 + h^2})\) Now, divide the cone surface area by the cylinder surface area: \(\frac{A_{cone}}{A_{cylinder}} = \frac{\pi r (\sqrt{r^2 + h^2})}{2\pi r h} = \frac{\sqrt{r^2 + h^2}}{2h}\) Since the fraction \(\frac{A_{cone}}{A_{cylinder}}\) is not a constant and depends on \(r\) and \(h\), the surface area of a cone does not equal one-third the surface area of a cylinder with the same radius and height. The relationship between the surface areas is given by: \(\frac{A_{cone}}{A_{cylinder}} = \frac{\sqrt{r^2 + h^2}}{2h}\)[