Question 333636
{{{528=3h^2+8h}}} Start with the given equation.



{{{0=3h^2+8h-528}}} Subtract 528 from both sides.



Notice that the quadratic {{{3h^2+8h-528}}} is in the form of {{{Ah^2+Bh+C}}} where {{{A=3}}}, {{{B=8}}}, and {{{C=-528}}}



Let's use the quadratic formula to solve for "h":



{{{h = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{h = (-(8) +- sqrt( (8)^2-4(3)(-528) ))/(2(3))}}} Plug in  {{{A=3}}}, {{{B=8}}}, and {{{C=-528}}}



{{{h = (-8 +- sqrt( 64-4(3)(-528) ))/(2(3))}}} Square {{{8}}} to get {{{64}}}. 



{{{h = (-8 +- sqrt( 64--6336 ))/(2(3))}}} Multiply {{{4(3)(-528)}}} to get {{{-6336}}}



{{{h = (-8 +- sqrt( 64+6336 ))/(2(3))}}} Rewrite {{{sqrt(64--6336)}}} as {{{sqrt(64+6336)}}}



{{{h = (-8 +- sqrt( 6400 ))/(2(3))}}} Add {{{64}}} to {{{6336}}} to get {{{6400}}}



{{{h = (-8 +- sqrt( 6400 ))/(6)}}} Multiply {{{2}}} and {{{3}}} to get {{{6}}}. 



{{{h = (-8 +- 80)/(6)}}} Take the square root of {{{6400}}} to get {{{80}}}. 



{{{h = (-8 + 80)/(6)}}} or {{{h = (-8 - 80)/(6)}}} Break up the expression. 



{{{h = (72)/(6)}}} or {{{h =  (-88)/(6)}}} Combine like terms. 



{{{h = 12}}} or {{{h = -44/3}}} Simplify. 



So the solutions are {{{h = 12}}} or {{{h = -44/3}}} 

  

However, a negative height doesn't make any sense. So we'll ignore {{{h = -44/3}}}



This means that the height is 12 mm.



I'll let you use this info to find the perimeter.