Question 1175766
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Let s be the side length of the equilateral triangle which is the base of the pyramid.<br>
The given slant height of 9 is the hypotenuse of a right triangle in which the legs are the height h of the pyramid and a segment from the center of the triangular base to the middle of one edge.  The segment from the center of the base to the middle of one edge is 1/3 of the altitude of the triangular base, which is sqrt(3)/2 times the side length s.  That makes the length of that segment {{{s*sqrt(3)/6}}}.<br>
The Pythagorean Theorem with that right triangle then gives us the equation<br>
{{{9^2 = h^2+(s*sqrt(3)/6)^2}}}
{{{81 = h^2+s^2/12}}}  [1]<br>
The volume of the pyramid is given as 50.<br>
{{{V = (1/3)Bh}}}<br>
where B is the area of the base.  The base is an equilateral triangle with side length s; its area is {{{s^2*sqrt(3)/4}}}.  So<br>
{{{V = (1/3)(s^2*sqrt(3)/4)(h)}}}<br>
So now we have the equation<br>
{{{50 = (1/3)(s^2*sqrt(3)/4)(h)}}}  [2]<br>
s is a parameter we are using in our analysis; our objective is to find the height, h.  To do that, we need to eliminate s (or, actually, s^2) between the two equations we have.<br>
Solve [2] for s^2... <br>
{{{50 = (1/3)(s^2*sqrt(3)/4)(h)}}}
{{{s^2 = 600/sqrt(3)(h) = 200*sqrt(3)(h)}}}  [3]<br>
... and substitute in [1]<br>
{{{81 = h^2+(50/3)*sqrt(3)(h)}}}  [4]<br>
That's the ugly part of the analysis. I leave the rest to you.<br>
(1) Use a graphing calculator or some other utility to solve equation [4] for the height h.
(2) Use that value of h in [3] to find the side length s.
(3) The lateral surface area is the area of three congruent isosceles triangles each with height 9 and base length s.<br>
Note you can verify your solution by seeing that the values you find for h and s satisfy equation [1].<br>