Answer to Question #270263 in Calculus for mimi

Question #270263

A manufacturer knows that if x goods are demanded on a particular week, the total cost and revenue functions will be:𝐢(π‘₯) = 14 + 3π‘₯ π‘Žπ‘›π‘‘ 𝑅(π‘₯) = 18π‘₯ βˆ’ 2π‘₯ 2 respectively.

i. Calculate the level of demand that will maximize profits (6 marks)

ii. Calculate the amount of profit that will be realized at this maximum point. (4 marks) 


1
Expert's answer
2021-11-24T02:33:39-0500

i)

profit function:

P(x)=𝑅(π‘₯)βˆ’πΆ(π‘₯)=18xβˆ’2x2βˆ’14βˆ’3x=βˆ’2x2+15xβˆ’14P(x)=𝑅(π‘₯)-𝐢(π‘₯)=18x-2x^2-14-3x=-2x^2+15x-14

Pβ€²(x)=βˆ’4x+15=0P'(x)=-4x+15=0

level of demand that will maximize profits:

x=15/4=3.75x=15/4=3.75


ii)

amount of profit that will be realized at this maximum point:

Pmax(x)=βˆ’2β‹…3.752+15β‹…3.75βˆ’14=14.125P_{max}(x)=-2\cdot3.75^2+15\cdot3.75-14=14.125


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