Answer:
[tex]54\pi\ cm^2[/tex]
Step-by-step explanation:
1. Approach
In order to solve this problem, one needs to find the height of the cone. This can be done using the Pythagorean theorem. After finding the height of the cone, one can use the formula to find the surface area of a cone, to find the surface area of this particular cone.
2. Find the height of the cone
The Pythagorean theorem is a property that relates the sides of a right triangle. The radius (distance from the center to circumference (outer edge)) of the base, the height of the cone, and the slant of the cone forms a right triangle. Thus, one can use the Pythagorean theorem to solve for the height. The Pythagorean states the following:
[tex]a^2+b^2=c^2[/tex]
Please note that (a) and (b) represent the sides adjacent (next to) the right angle, (c) represents the hypotenuse (the side opposite the right angle). In this case (a) is the height, and (b) is the radius of the base. (c) is the diagonal of the cone. Substitute the given values in and solve for the height,
[tex]a^2+b^2=c^2[/tex]
[tex]a^2+3^3=15^2[/tex]
Simplify,
[tex]a^2+3^3=15^2[/tex]
[tex]a^2+9=225[/tex]
Inverse operations,
[tex]a^2+9=225[/tex]
[tex]a^2=216[/tex]
[tex]a=\sqrt{216}[/tex]
3. Find the surface area of the cone
The following formula can be used to find the surface area of a cone,
[tex]A_s=\pi r(r+\sqrt{h^2+r^2})[/tex]
Where (h) is the height of the cone, (r) is the radius of the base of the cone, and ([tex]\pi[/tex]) represents the value (3.1415...). Substitute the given values into the formula and solve for the surface area,
[tex]A_s=\pi r(r+\sqrt{h^2+r^2})[/tex]
[tex]A_s=\pi(3)(3}+\sqrt{\sqrt{216}^2+3^2}[/tex]
Simplify,
[tex]A_s=\pi(3)(3}+\sqrt{\sqrt{216}^2+3^2}[/tex]
[tex]A_s=\pi(3)(3}+\sqrt{216+9})[/tex]
[tex]A_s=\pi(3)(3}+\sqrt{225})[/tex]
[tex]A_s=\pi(3)(3}+15)[/tex]
[tex]A_s=\pi(3)(18)[/tex]
[tex]A_s=\pi(54)[/tex]
