Questions: Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p=160-0.75x. It also determines that the total cost of producing x suits is given by C(x)=4000+0.5x^2. a) Find the total revenue, R(x). b) Find the total profit, P(x). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?

Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p=160-0.75x. It also determines that the total cost of producing x suits is given by C(x)=4000+0.5x^2. a) Find the total revenue, R(x). b) Find the total profit, P(x). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?
Transcript text: Raggs, Ltd. a clothing firm, determines that in order to sell $x$ suits, the price per suit must be $p=160-0.75 \mathrm{x}$. It also determines that the total cost of producing $x$ suits is given by $\mathrm{C}(\mathrm{x})=4000+0.5 \mathrm{x}^{2}$. a) Find the total revenue, $\mathrm{R}(\mathrm{x})$. b) Find the total profit, $\mathrm{P}(\mathrm{x})$. c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?
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Solution

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Solution Steps

Step 1: Calculate the total revenue function

Total revenue function, $R(x) = x \cdot p(x) = x \left(160 - 0.75 x\right)$.

Step 2: Determine the total profit function

Total profit function, $P(x) = R(x) - C(x) = - 0.5 x^{2} + x \left(160 - 0.75 x\right) - 4000$.

Step 3: Find the derivative of $P(x)$ with respect to $x$, $P'(x)$

$P'(x) = 160 - 2.5 x$.

Step 4: Determine the value of $x$ that maximizes $P(x)$

The critical point(s) for maximum profit is/are [64]. After evaluating the second derivative, the value of $x$ that maximizes profit is 64.

Step 5: Calculate the maximum profit

The maximum profit is $1120$.

Step 6: Determine the price per unit at this level of output

The price per unit at this level of output is $112$.

Final Answer:

The maximum profit is achieved by producing and selling 64 units, resulting in a profit of $1120$ at a price per unit of $112$.

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