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

If not, what is the correct relationship between the two surface areas? Answer: No, the lateral surface area of a cone of radius 'r' and height 'h' is not equal to one-third the lateral surface area of a cylinder with the same radius and height. The correct relationship between the two surface areas is: LSA_cone = (l / (2h)) * LSA_cylinder, where 'l' is the slant height of the cone.

Step 1: 1. Lateral Surface Area of a Cone

: The formula for the lateral surface area (LSA) of a cone is given as: LSA_cone = πr * l, where 'r' is the radius of the base, and 'l' is the slant height of the cone. To calculate the value of 'l', we use Pythagorean theorem as follows: l^2 = r^2 + h^2, l = sqrt(r^2 + h^2).

Step 2: 2. Lateral Surface Area of a Cylinder

: The formula for the lateral surface area (LSA) of a cylinder is given as: LSA_cylinder = 2πrh, where 'r' is the radius of the base, and 'h' is the height of the cylinder.

Step 3: 3. Establish the relationship between the surface areas

: Now, let's check whether LSA_cone is one-third of LSA_cylinder. LSA_cone = (1/3) * LSA_cylinder => πr * l = (1/3) * 2πrh => l = (2/3) * h. However, we know that l = sqrt(r^2 + h^2), and by squaring both sides, l^2 = r^2 + h^2. Substitute l = (2/3) * h in the above equation, (4/9) * h^2 = r^2 + h^2. As we can see, this equation doesn't have a general solution for all possible values of 'r' and 'h', which means that the surface area of a cone of radius 'r' and height 'h' is not equal to one-third the surface area of a cylinder with the same radius and height. To find the correct relationship, we need to compare the surface areas directly.

Step 4: 4. Correct Relationship between the Surface Areas

: Given LSA_cone = πr * l and LSA_cylinder = 2πrh, we need to find the relationship between the two surface areas. Divide LSA_cone by LSA_cylinder and simplify: LSA_cone / LSA_cylinder = (πr * l) / (2πrh) = l / (2h). This tells us that the lateral surface area of a cone is equivalent to l/(2h) times the lateral surface area of a cylinder—so long as the radius and the height of both the cone and the cylinder are the same.