The profit P, in thousands of dollars, that a manufacturer makes is a function of the...

Free

50.1K

Verified Solution

Question

Basic Math

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

Answer & Explanation Solved by verified expert

3.6 Ratings (579 Votes)

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

-----------------------------------------

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

-----------------------------------------

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

--------------------------------------------

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

Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:

Unlimited Question Access with detailed Answers
Zin AI - 3 Million Words
10 Dall-E 3 Images
20 Plot Generations
Conversation with Dialogue Memory
No Ads, Ever!
Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!

Become a Member