Answer:
[tex]\approx19.16[/tex]
Step-by-step explanation:
To find the surface area of a cone, one uses the following formula,
[tex]A=(\pi)(r)(r+\sqrt{h^2+r^2}})[/tex]
Where (r) is the radius of the base of the cone, and (h) is the height of the cone, ([tex]\pi[/tex]) represents the numerical constant (3.1415...). In order for this formula to work, one needs the height of the cone. This can be found using the Pythagorean theorem. One can form a right triangle with the base's radius, the side length, and height of the cone, then solve for the height of the cone. The Pythagorean theorem states the following,
[tex]a^2+b^2=c^2[/tex]
Substitute,
[tex]a^2+(1)^2=(5.1)^2[/tex]
Simplify,
[tex]a^2+1=26.01[/tex]
Inverse operations,
[tex]a^2=25.01\\a\approx5[/tex]
Now one can use the formula to find the surface area of a cone, substitute in the given values, and solve,
[tex]A=(\pi)(r)(r+\sqrt{h^2+r^2}})[/tex]
Substitute,
[tex]A=(\pi)(1)(1+\sqrt{5^2+1^2}})[/tex]
[tex]A=(\pi)(1+\sqrt{25+1})\\A=\pi(1+\sqrt{26})\\A\approx 19.16[/tex]
