<PRE><B>Question 1116879</B>
&lt;br&gt;
Let the altitude of the pyramid be h.  Then consider the right triangle with that altitude as one leg and the slant height of 9 as the hypotenuse.  Then the other leg of that right triangle has length {{{sqrt(81-h^2)}}}.&lt;br&gt;
That other leg of the right triangle is one-third the length of the altitude of the triangular base; the length of that altitude is {{{sqrt(3)/2}}} times the side length s of the triangular base.  So&lt;br&gt;
{{{s(sqrt(3)/2) = 3(sqrt(81-h^2))}}}&lt;br&gt;
{{{s = (2*sqrt(3))(sqrt(81-h^2))}}}&lt;br&gt;
We can find the height of the pyramid by knowing that its volume is 50:&lt;br&gt;
volume = one-third base times height&lt;br&gt;
{{{50 = (1/3)((s^2)(sqrt(3)/4))(h)}}}&lt;br&gt;
{{{50 = (1/3)((12(81-h^2))(sqrt(3)/4))(h)}}}&lt;br&gt;
{{{50 = (81-h^2)(h)(sqrt(3))}}}&lt;br&gt;
A graphing calculator shows two solutions to that equation; to several decimal places the solutions are h=0.35695 and h=8.81621.&lt;br&gt;
So there will be two different pyramids that give the correct volume.  The one with the very small height is probably not the intended one, so from here I will assume the height is 8.81621.&lt;br&gt;
Substituting that value in the expression for the side length of the triangular base,&lt;br&gt;
{{{s = (2*sqrt(3))(sqrt(81-8.81621^2)) = 6.26844}}}&lt;br&gt;
Now the lateral surface area consists of three congruent triangles, each with base 6.26822 and height 8.81621.  The total surface area is&lt;br&gt;
{{{(3)(1/2)(6.26822)(8.81621) = 84.62389}}}&lt;br&gt;
This is very close to the given answer.  Since I kept 5 or more decimal places in all my calculations, I suspect that the given answer 84.65 was obtained keeping fewer decimal places in the calculations.</PRE>