/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 The volume of a cone of radius \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 30

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.

Short Answer

Expert verified
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 by step solution

01

1. Lateral Surface Area of a Cone

: The formula for the lateral surface area (LSA) of a cone is given as: LSA_cone = 蟺r * 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).
02

2. Lateral Surface Area of a Cylinder

: The formula for the lateral surface area (LSA) of a cylinder is given as: LSA_cylinder = 2蟺rh, where 'r' is the radius of the base, and 'h' is the height of the cylinder.
03

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 => 蟺r * l = (1/3) * 2蟺rh => 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.
04

4. Correct Relationship between the Surface Areas

: Given LSA_cone = 蟺r * l and LSA_cylinder = 2蟺rh, we need to find the relationship between the two surface areas. Divide LSA_cone by LSA_cylinder and simplify: LSA_cone / LSA_cylinder = (蟺r * l) / (2蟺rh) = 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鈥攕o long as the radius and the height of both the cone and the cylinder are the same.

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