Sphere into a cone!

The volume of a sphere is given by $V_{s}=\dfrac{4}{3}\pi r^3$ so the volume of the sphere is $\dfrac{4}{3} \pi (7^3)$ . The volume of a cone is given by $V_{c}=\dfrac{1}{3} (\pi r^2)(h)$ , so the volume of the cone is $\dfrac{1}{3} \pi (7^2)(h)$ . Since the two volumes are equal, we have

$\dfrac{4}{3} \pi (7^3)= \dfrac{1}{3} \pi (7^2)(h)$

$h=28$

It follows that slant height is $L=\sqrt{28^2+7^2}=7\sqrt{17}$ .

Now the surface area of the cone is equal to the lateral area plus the area of the base, and it is given by $A=\dfrac{1}{2}CL + \pi r^2$ where $C$ is the circumference of the base and $L$ is the slant height. SO the desired answer is

$A=\dfrac{1}{2}(2)(\pi)(7)(7\sqrt{17})+\pi (7^2) \approx \color{#D61F06}\boxed{788.64}$