Consider the following table that describes the total cost of producing dresses by Shirley, a local fashion artist. Dresses Total cost ($) 0 50 1 80 2 105 3 140 4 180 5 225 6 275 7 350 8 440 Assume that the market is competitive and the price of dresses is $64. At this price how many dresses should Shirley produce to maximise profit? (If Shirley is indifferent between producing an additional dress and not producing, please assume that she will produce the additional dress to break the tie.)

To find out the optimal number of dresses Shirley should produce to maximize profit, we can follow these steps:

1. Calculate the Total Revenue (TR) at different quantities.

2. Calculate the Total Cost (TC) at these quantities.

3. Calculate the Profit, which is the difference between Total Revenue and Total Cost.

4. Find the quantity of dresses that yields the maximum profit.

Given:

- The price per dress is $64.

- Total costs are given for producing up to 8 dresses.

The total revenue for producing and selling \( Q \) dresses is given by:

TR = P \times Q

where \( P \) is the price per dress ($64).

The profit for producing \( Q \) dresses is:

\text{Profit} = TR - TC

Here's the detailed calculation:

[SOLUTION]

\begin{array}{c|c|c|c}\text{Dresses (Q)} & \text{Total Revenue (TR, \$)} & \text{Total Cost (TC, \$)} & \text{Profit (TR - TC, \$)} \\\hline0 & 0 \times 64 = 0 & 50 & 0 - 50 = -50 \\1 & 1 \times 64 = 64 & 80 & 64 - 80 = -16 \\2 & 2 \times 64 = 128 & 105 & 128 - 105 = 23 \\3 & 3 \times 64 = 192 & 140 & 192 - 140 = 52 \\4 & 4 \times 64 = 256 & 180 & 256 - 180 = 76 \\5 & 5 \times 64 = 320 & 225 & 320 - 225 = 95 \\6 & 6 \times 64 = 384 & 275 & 384 - 275 = 109 \\7 & 7 \times 64 = 448 & 350 & 448 - 350 = 98 \\8 & 8 \times 64 = 512 & 440 & 512 - 440 = 72 \\\end{array}

Therefore, Shirley should produce 6 dresses to maximize her profit, which is $109.

[STEP-BY-STEP]

1. Calculate the total revenue for each quantity (TR = P * Q).

\begin{array}{c|c}\text{Dresses \lparen Q\rparen} & \text{Total Revenue \lparen TR, \$\rparen} \\ 0 & 0\times64=0 \\ 1 & 1\times64=64 \\ 2 & 2\times64=128 \\ 3 & 3\times64=192 \\ 4 & 4\times64=256 \\ 5 & 5\times64=320 \\ 6 & 6\times64=384 \\ 7 & 7\times64=448 \\ 8 & 8\times64=512 \\ & \placeholder{}\end{array}

2. Subtract the Total Cost from Total Revenue to find the profit for each quantity.

\begin{array}{c|c}\text{Dresses \lparen Q\rparen} & \text{Profit \lparen TR - TC, \$\rparen} \\ 0 & 0-50=-50 \\ 1 & 64-80=-16 \\ 2 & 128-105=23 \\ 3 & 192-140=52 \\ 4 & 256-180=76 \\ 5 & 320-225=95 \\ 6 & 384-275=109 \\ 7 & 448-350=98 \\ 8 & 512-440=72 \\ & \placeholder{}\end{array}

3. Identify the maximum profit and the corresponding quantity.

\text{Maximum profit is 109,}

\text{which is earned by producing 6 dresses.}

Thus, Shirley should produce 6 dresses to maximize her profit.