It is given that the diameter of the base of the cone is AB=20, and the slant height is BC=10.
Recall that the radius is one-half of the diameter, it follows that:
[tex]r=\frac{1}{2}\cdot20=10[/tex]
Note that the radius (r), vertical height (h), and slant height (l) of a cone form a right triangle which satisfies the Pythagorean Theorem:
[tex]h^2+r^2=l^2[/tex]
Substitute r=10 and l=10 into the equation:
[tex]\begin{gathered} h^2+10^2=10^2 \\ \Rightarrow h^2=10^2-10^2 \\ \Rightarrow h^2=0 \\ \Rightarrow h=0 \end{gathered}[/tex]
Notice that the height of the cone is calculated to be zero which is not possible for a cone.
The Surface Area of a cone is:
[tex]S=\pi r^2+\pi rl[/tex]
Substitute r=10 and l=10 into the formula:
[tex]\begin{gathered} S=\pi(10^2)+\pi(10)(10) \\ \Rightarrow S=100\pi+100\pi=200\pi\text{ square units} \end{gathered}[/tex]
The answer is 200π square units.
