Exercise 29 Page 648

The given solid is a cone with a diameter of $4.5$ centimeters and a height of $12$ centimeters.

To calculate the surface area of a cone, we can use the known formula where $r$ is the radius of the base and $\ell$ is the slant height of the cone. Looking at the diagram, we know that the diameter of the cone's base is $4.5$ centimeters. By dividing by $2,$ we get the radius of the base. We now know that the radius of the cone is $2.25$ centimeters and that the height of the cone is $12$ centimeters. Now, we can focus on the right triangle created within the cone. The slant height of the cone is the hypotenuse of the triangle. The legs of the triangle are the height of the pyramid $h$ and the radius $r$ of the base. Notice that we can find the slant height of the cone using the Pythagorean Theorem. Let's substitute $12$ for $h$ and $2.25$ for $r$ into the above equation to find the slant height of the cone. We found that the slant height is about $12.2$ centimeters. Note that, when solving the above equation, we only kept the principal root because $\ell$ represents a side length and must be a positive number. Let's now substitute $r$ with $2.25$ and $\ell$ with $12.2$ into the formula, we can calculate $S.$ The surface area of the cone is approximately $102.1$ square centimeters.