Question 1150480
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(a) Find an expression for the volume in terms of x.<br>
The two triangular bases have side lengths 5x, 12x, and 13x.<br>
The sum of the lengths of the sides of the two bases is 60x.<br>
The sum of the lengths of all the edges of the prism -- the sides of the bases, plus the three edges connecting the two bases -- is 180.<br>
Let h be the height of the prism; then<br>
{{{60x+3h = 180}}}
{{{3h = 180-60x}}}
{{{h = 60-20x}}}<br>
The volume is the area of the base, times the height:<br>
{{{V = ((1/2)(12x)(5x))(60-20x)}}}
{{{V = 30x^2(60-20x) = 1800x^2-600x^3}}}<br>
(b) Find the value of x that maximizes the volume.  (Find the value of x for which the derivative is zero.)<br>
{{{dV/dx = 3600x-1800x^2}}}
{{{3600x-1800x^2 = 0}}}
{{{1800x(2-x) = 0}}}
{{{x = 2}}}<br>
Note x=0 also makes the derivative zero but makes no sense in the actual problem.<br>