Price function: p(x)=150-0.5x and cost function: c(x)=4000+0.25x^2 How do u you find total rev. and max profit

I need to know total revenue, total profit,how much (X) needs to be produced to maximize profit, what is the maximum profit, and what price per X that must be charged in order to make this maximum profit.

Price function: p(x)=150-0.5x and cost function: c(x)=4000+0.25x^2 How do u you find total rev. and max profit
Revenue = p*x = (150-0.5x)x = 150x - 0.5x^2





Profit = Revenue - Cost


= 150x - x^2 - (4000+0.25x^2)


= -(5/4)x^2 + 150x - 4000





To find how much x needs to be produced to maximize the profit, you need to find the vertex.


The first coordinate represents x.


The second number represents the maximum profit.


To find the price, plug in the x-value you found.





See the website below for help with finding the vertex.
Reply:x = # of items that are produced and sold





Total Revenue = r(x) = x * p(x) = 150x - 0.5x^2





Total Profit = Total Revenue - Cost = P(x) = r(x) - c(x)





= (150x - 0.5x^2) - (4000 + 0.25x^2)





= -0.75x^2 + 150x - 4000





The graph of the profit function is a downward facing parabola, which means that it has a maximum value (the vertex) at some value of x. Accordingly, the value of the function's first derivative (i.e. the tangent line to the function) will be zero at this value.





dP(x)/dx = -1.5x + 150 = 0





1.5x = 150 ----------%26gt; x = 100





*100 product units must be sold to acheive the max profit*





If x = 100, then the max profit amount is:





P(100) = -0.75*100^2 + 150*100 - 4000





= -7500 + 15000 - 4000





= $3500





Selling price needed to achieve max profit = p(100)





p(100) = 150 - 0.5*100 = 150 - 50 = $100

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