Question 1195270

 the demand price for selling out a production run of {{{x}}} DVDs is given by 

{{{p(x)=-0.0005x^2+60}}}  


the weekly cost of producing  is given by 

{{{C(x)=-0.001x^2+18x+4000 }}} 

so, we have

demand price: {{{ p(x)=-0.0005x^2+60 }}}
weekly cost : {{{C(x)=-0.001x^2+18x+4000}}}


i)

the Revenue function {{{R(x)}}}:

Revenue is equal to the number of units sold times the price per unit. To obtain the revenue function, multiply the output level by the price function.


{{{R(x)= x*p(x)}}}

{{{R(x)= x*(-0.0005x^2+60)}}}

{{{R(x)= -0.0005x^3+60x}}}


DOMAIN:  {{{R }}}(all real numbers)
 
so, domain is all {{{x >=0}}}

interval notation: 

[{{{0}}}, {{{infinity}}})


ii)


The profit a business makes is equal to the revenue it takes in minus what it spends as costs. To obtain the profit function, subtract costs from revenue.


{{{P(x)=R(x) - C(x)}}}

{{{P(x)=-0.0005x^3+60x - (-0.001x^2+18x+4000)}}}

{{{P(x)=-0.0005x^3+60x +0.001x^2-18x-4000}}}

{{{P(x)=-0.0005x^3 + 0.001x^2 + 42x - 4000}}}


DOMAIN:  {{{R}}} (all real numbers)


so, domain is all {{{x >=0}}}

interval notation: 

[{{{0}}}, {{{infinity}}})