Math 241
AJ-773Math 241 – Activity
A stay at home mom wants to supplement her home income by making candles and selling them at the local craft mall. She has asked you to help her out to find a price and maximize her revenue for her first month. Your goal is to help her optimize her sales.
She will sell candles that she will make herself in her garage. Each candle uses 4 oz. of 25
dye, 1 wick (6 inches in length), 1 pound of wax, and 1⁄2 oz. of scent. The materials prices are listed below:
She bought a $50 roaster to melt the wax in, 15 molds at $8.95 each, and a wax pouring pot for $10.95. She is going to sell the candles at the local craft mall. The craft mall charges $35 a month to rent a 1 foot by 2 foot space. They will also charge her 50¢ per candle sold.
To help decide pricing she asked the craft mall for some sales results for other candle sellers. She got the following monthly data:
Find each of the following functions and perform the appropriate mathematics to help maximize the candle seller’s profits. Write up the information that you find in a report and attach all supporting mathematics to the report. (Round all decimals to two places.)
1. Find the cost of the supplies per candle. Use this information to find the total cost function, C(q), for her to sell in the craft mall for one month? (Remember to factor in ALL costs, not just the craft mall costs and do not factor in left over material. Use the information in the table to find the “per candle” price for each material. Also, q = the quantity of candles produced.)
2. Find a linear function for demand using linear regression. (You may use your calculator or Excel.) Remember with demand, x = p = price, and y = q = demand. Make sure your function is written in terms of (p , q) instead of (x , y).
3. In question 1 you found the function for cost per unit produced or q. Now use the answers from questions 1 and 2 to find the function C(p).
4. What is the revenue function, R(p), for her for the first month? 5. What is the profit function, P(p), for her for the first month?
Item | Price |
Dye | $13.95 for 4oz. |
Wicks | $5.95 for 100 6 in. wicks |
Wax | $55.00 for a 50lb case |
Scent | $16.95 for 16oz. |
Price, p | $15 | $13.75 | $12.50 | $10.25 |
Quantity Sold, q | 47 | 68 | 81 | 104 |
6. What price will yield the maximum profit? What is that maximum profit? How many candles will she need to supply to maximize the profit? (Round your answers appropriately.)
7. Use the equation you found in question number 1 to find the Average Cost function, C (q) = C(q) . Use this equation to find the average cost of the candles when you are
q
producing the amount of candles that will maximize the profit.
8. Find the marginal cost,C′(p), marginal revenue, R′(p), and marginal profit, P′(p), functions. Discuss what each equation means for this situation and in particular the result
from plugging in the price that will give the maximum profit.
9. In the above problem we just discussed the profit for the first month of operation. Do you think that she should use the same cost function every month? What would be the cost function for the second month, or for the third month? Would this effect what she should charge for her candles?
Write Up: Your report should include a full explanation of what you did in this activity, how you arrived at your mathematical results and justification of your conclusions. Answer all questions posed along the way. The write up should be in the form of a report and not a list of numbered answers to questions. You can include the supporting math in the report or you can refer to it and attach it at the end. You may work in groups, but you each must hand in your own paper. It does not need to be typed, but it needs to be legible.
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