A commodity has a demand function modeled by p=126−0.5x and a to

Sheelmgal1p 2021-11-22 Answered
A commodity has a demand function modeled by \(\displaystyle{p}={126}−{0.5}{x}\) and a total cost function modeled by \(\displaystyle{C}={50}{x}+{39.75}\), where x is the number of units.
(a)Use the first marginal analysis criterion presented in class to find the production level that yields a maximum profit. Then use the second criterion to ensure that level yields a maximum and not a minimum. Finally, what unit price (in dollars) yields a maximum profit?
$ per unit
(b)When the profit is maximized, what is the average total cost (in dollars) per unit? (Round your answer to two decimal places.)
$ per unit

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Expert Answer

Blanche McClain
Answered 2021-11-23 Author has 6977 answers

Step 1
Demand function \(\displaystyle{P}={126}-{0.5}{x}{\left({1}\right)}\)
Total cost function \(\displaystyle{C}={50}{x}+{39.75}\)
a). Revenue R(x) =price x number sold
\(\displaystyle{R}{\left({x}\right)}={P}{x}\)
\(\displaystyle{R}{\left({x}\right)}={\left({126}-{0.5}{x}\right)}{x}\)
\(\displaystyle{R}{\left({x}\right)}={126}{x}-{0.5}{x}^{{{2}}}\)
Step 2
Profit P(x)=revenue-cost
\(\displaystyle={\left({126}{x}-{0.5}{x}^{{{2}}}\right)}-{\left({50}{x}+{39.75}\right)}\)
\(\displaystyle={126}{x}-{0.5}{x}^{{{2}}}-{50}{x}-{39.75}\)
\(\displaystyle=-{0.5}{x}^{{{2}}}+{76}{x}-{39.75}\)
To maximize profit P'(x)=0
\(\displaystyle{P}'{\left({x}\right)}=-{0.5}{\left({2}{x}\right)}+{76}\)
\(\displaystyle{P}'{\left({x}\right)}=-{x}+{76}={0}\)
x=76
and \(\displaystyle{P}{''''}{\left({x}\right)}=-{1}{<}{0}\) (maximum)
first and second derivative shows that at x=76, P(x) is maximum.

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Huses1969
Answered 2021-11-24 Author has 7160 answers
Step 1
b
Average cost \(\displaystyle\overline{{{C}}}{\left({x}\right)}={\frac{{{C}{\left({x}\right)}}}{{{x}}}}={\frac{{{50}{x}+{39.75}}}{{{x}}}}\)
substitute the value of x=76, we get
\(\displaystyle\overline{{{C}}}{\left({x}\right)}=\overline{{{C}}}{\left({76}\right)}={\frac{{{50}{\left({76}\right)}+{39.75}}}{{{76}}}}\)
\(\displaystyle\overline{{{C}}}{\left({x}\right)}=\${50.52}\) per unit
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