Question 959010

Woods Limited manufactures and sells {{{x}}} small patio tables each day. 
The daily cost of production is modeled as 
{{{C(x) = 3x^2 + 1500}}}, and the revenue produced each day is modeled as {{{R(x) = 180x}}}. 
Suppose the company is limited by time to producing at least {{{30}}} (at least means ≥{{{30}}})tables every day.
 
1. What is the break-even point? 
to find it set {{{R(x) = C(x)}}}, then solve the resulting quadratic equation  
he break-even point occurs when 
{{{ 3x^2 + 1500=180x}}} 
 {{{ 3x^2 -180x+ 1500=0}}} .......simplify, divide all terms by {{{3}}}
{{{ x^2 -60x+ 500=0}}} ........use quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{x = (-(-60)+- sqrt( (-60)^2-4*1*500 ))/(2*1) }}} 

{{{x = (60+- sqrt( 3600-2000 ))/2 }}} 

{{{x = (60+- sqrt( 1600 ))/2 }}} 

{{{x = (60+- 40)/2 }}}

solutions:

{{{x = (60+ 40)/2 }}}

{{{x = 50 }}}

or

{{{x = (60- 40)/2 }}}

{{{x = 10 }}}

your solution is {{{highlight(x=50)}}} because the company is producing at least {{{30}}} tables to have the break-even point ; so, disregard {{{x = 10 }}}


2. If the company sells {{{45}}} tables, then what is the profit? 

 The profit function {{{P(x) = R(x) - C(x)}}}, 

write {{{P(x)}}} in terms of {{{x}}}, then evaluate {{{P(45)}}}

{{{P(x) = 180x -3x^2 + 1500}}}

{{{P(45) = 180*45 -3*45^2 + 1500}}}  

{{{P(45) = 8100 -6075 + 1500}}}  

{{{P(45) = 3525}}}=> the profit the company earns if sells {{{45}}} tables


3. 

How many tables must the company produce in order to get a profit of ${{{1125}}}? 

Set {{{P(x) = 1125}}}, then solve the resulting quadratic equation

{{{1125 = 180x -3x^2 + 1500}}}

{{{3x^2-180x+1125-1500=0}}}

{{{3x^2-180x-375=0}}}

{{{x^2-60x-125=0}}}........use quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{x = (-(-60) +- sqrt( (-60)^2-4*1*(-125) ))/(2*1) }}} 

{{{x = (60 +- sqrt( 3600+500 ))/2 }}} 


{{{x = (60 +- sqrt( 4100 ))/2 }}} 

{{{x = (60 +- 64.03124237432849)/2 }}}

{{{x = (60 +- 64.03)/2 }}}

we need only positive value since {{{x}}} represents an order

{{{x = (60 + 64.03)/2 }}}

{{{x = 124.03/2 }}} 

{{{x = 62.01562118716424 }}}- exact solution

{{{x = 62 }}}round it to whole number since {{{x}}} represents tables




4. 

What happen if the company produces {{{58}}} tables? 

evaluate {{{P(58)}}}

{{{P(58) = 180*58 -3*58^2 + 1500}}} 

{{{P(58) = 10440 -10092 + 1500}}} 

{{{P(58) = 348 + 1500}}}

{{{P(58) = 1848}}}  => the profit the company earns if sells {{{58}}} tables