Question 402399
{{{P(x) = R(x) - C(x)}}}
{{{P(x) = 90x - x^2 - (200 + 60x)}}}
{{{P(x) = 90x - x^2 - 200 - 60x}}}
(a)
{{{P(x) = -x^2 + 30x - 200}}}
(b)
{{{0 = -x^2 + 30x - 200}}}
use the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{a = -1}}}
{{{b = 30}}}
{{{c = -200}}} 
{{{x = (-30 +- sqrt( 30^2-4*(-1)*(-200) ))/(2*(-1)) }}} 
{{{x = (-30 +- sqrt( 900 - 800 ))/(-2) }}} 
{{{x = (-30 +- sqrt( 100 ))/(-2) }}} 
{{{x = (-30 + 10)/(-2) }}} 
{{{x = 10}}}
and
{{{x = (-30 - 10)/(-2) }}} 
{{{x = 20}}}
there is zero profit for selling 10  or 20 snowblowers
(c)
For selling 18 snowblowers:
{{{P(x) = -x^2 + 30x - 200}}}
{{{P(18) = -18^2 + 30*18 - 200}}}
{{{P(18) = -324 + 540 - 200}}}
{{{P(18) = 16}}}
There is $16K profit for selling 18 snowblowers
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Note that you can see the whole picture by plotting {{{P(x)}}}
{{{ graph( 300, 300, -5, 30, -5, 30, -x^2 + 30x - 200) }}}
Note that there is no profit for selling 10 or less snowblowers
or 20 or more snowblowers