Exercise 26 Page 637

We want to find the slant height and the lateral area of the following conical hat.

Let's start by finding the slant height. To do so, we can use the fact the slant height of a cone makes a right triangle with the height and the radius.

Notice that we know the lengths of the legs of the triangle. To find the hypotenuse, we will use the Pythagorean Theorem. In the formula, $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse of a right triangle. We are given the triangle with $a=7$ and $b=14.$ Let's apply the Pythagorean Theorem to this triangle. Now we can solve an equation that we got to find $c.$ The slant height of the hat is about $15.7$ inches. Next, we will find the lateral area of the hat. Recall that the lateral area of a cone is equal to one-half the circumference of the base times the slant height. To calculate the lateral area of a cone, we can use the following formula. In this formula, $r$ is the radius of the base and $\ell$ is the slant height of the cone. We can substitute the slant height and the radius of the conical hat into the formula and calculate its lateral area. Let's do it! We got that the lateral area of the hat is about $345.3$ square inches.