Exercise 19 Page 872

Consider the given solid.

The given solid is a cone. It has a height of 22 centimeters and a diameter of 10 centimeters. The length of the slant height is unknown. Let's first calculate the lateral area and then the surface area.

Lateral Area

To calculate the lateral area of a cone, we can use the known formula where r is the radius of the base and l is the slant height. L=π rlFrom the diagram, we know that the diameter of the cone's base is 10. By dividing by 2, we get the radius of the base. r=10/2= 5 Therefore, the radius of the base is 5 centimeters.

Also, we can see a right triangle formed by the height of the cone, the radius of the base, and the slant height.

Let's use the Pythagorean theorem to find the slant height l.

By taking the positive square root of each side, we have that c=sqrt(509). Therefore, the slant height of the cone is sqrt(509).

Now, we have l= sqrt(509) and r= 5. Let's substitute these values into the formula for the lateral area and calculate L

The lateral area of the cone is about 354.4 square centimeters.

Surface Area

To calculate the surface area of a cone, we can use the known formula where r is the radius of the base and l is the slant height of the cone. S=π rl+π r^2 Earlier, we found that r= 5 and π rl ≈ 354.4. Let's substitute these values into the formula to calculate S.

The surface area of the cone is about 432.9 square centimeters.