Question

The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N, in thousands, of widgets produced in a year, and the formula is P(N)= 10N-N²-6.34. The formula is valid up to a level of 7,000 widgets produced. Express in functional notation the profit at a production level of 4,500 widgets. Calculate the value and explain the result in practical terms. What are the fixed costs? Determine the break-even point(s) for this manufacturer. What is the maximum profit?

Answer 1

given P(N)= 10N-N²-6.34

4,500 widgets
=>N=4.5

Expression for the profit at a production level of 4,500 widgets is P(4.5)

P(4.5)= 10(4.5)-4.5²-6.34
=>P(4.5)= 45-22.5-6.34
=>P(4.5)= 18.41

value is 18.41
profit at a production level of 4,500 widgets is 18.41 thousand dollars

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fixed cost =P(0)
=>fixed cost =10(0)-(0)²-6.34
=>fixed cost =-6.34 thousand dollars

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at breakeven profit is zero
=>10N-N²-6.34=0
=>N²-10N=-6.34
=>N²-2(5)N+5²=5²-6.34
=>(N-5)²=18.66
=>N-5=-(18.66)^1/2
=>N=5-(18.66)^1/2
=>N=0.6802777867.......thousand widgets

break-even point is 680 widgets

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P(N)= 10N-N²-6.34
=>P(N)= -(N²-10N)-6.34
=>P(N)= -(N²-2(5)N+5²-5²)-6.34
=>P(N)= -(N-5)²+5²-6.34
=>P(N)= -(N-5)²+18.66

maximum profit is 18.66 thousand dollars which occurs when 5000 widgets are produced in an year