Maximizing Profit: Balancing Cost and Price for 300 Units

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In summary, the conversation discusses a company's ability to produce 300 units per week at a cost of $58 per unit, with a charge of $100 per unit. It also explores the effect of a $5 price increase on the number of units sold, and the formula for calculating profit. The conversation concludes with the question of what type of rule the formula represents, with the answer being a product of two polynomials.
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atkinslm
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Homework Statement



A company can do 300 units per week at a weekly cost per unit of $58 for labor and supplies. If the charge is $100 per unit, then it would have clients for 300 units. However, for each $5 price increase, it could expect to have 10 fewer units.

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The Attempt at a Solution

 
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  • #2
Can you come up with a formula for profit?

Let's do the easy one first, when the company doesn't charge anything extra:
Pi(x) = (number of units)(sell price - cost) = 300(100-58)
Fairly straight forward, now let's modify this with a variable “x” which will represent how many times they will charge an extra $5.

P(x) = (number of units)(sell price - cost) = (300-10x)(100+5x-58) = (300-10x)(42+5x)
Now what we do from here depends on what class you are in. If you are in an algebra course, you should recognize this as a parabola and know how to graph it. If you are in calculus I would expand it, take the derivative, and set it equal to zero.
 
  • #3
Thanks for your help so far. I am in Precalculus. I need to plug these formulas into an excel spreadsheet.
 
  • #4
I have the formulas in the spread sheet. I think all is going well. I need to know what type of rule is this. I would really like to be able to figure this out. Is it the product of two polynomials?
 
  • #5
It is two polynomials multiplied. You should be able to do this by hand by using the FOIL method then putting it standard form for a parabola. Picture a parabola, what point would be the max or min?
 
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  • #6
Thanks alot!
 

FAQ: Maximizing Profit: Balancing Cost and Price for 300 Units

How can I find the optimal cost and price for 300 units to maximize profit?

To find the optimal cost and price for 300 units, you can use a mathematical formula called the profit maximization rule. This rule states that profit is maximized when marginal revenue (the change in revenue from selling one additional unit) is equal to marginal cost (the change in cost from producing one additional unit). By setting these two values equal to each other and solving for cost and price, you can determine the optimal values for maximizing profit.

What factors should be considered when balancing cost and price for 300 units?

When balancing cost and price for 300 units, several factors should be taken into account. These include fixed costs (such as rent and equipment), variable costs (such as materials and labor), market demand, competition, and desired profit margin. It is important to carefully analyze these factors to determine the most effective cost and price strategy for maximizing profit.

How can I reduce costs while maintaining a competitive price for 300 units?

Reducing costs while maintaining a competitive price for 300 units can be achieved through various strategies. These include improving production efficiency, negotiating better deals with suppliers, minimizing waste, and finding alternative materials or processes. It is also important to regularly review and adjust prices in accordance with market trends and competitor pricing to remain competitive.

Is it possible to maximize profit without compromising quality for 300 units?

Yes, it is possible to maximize profit without compromising quality for 300 units. In fact, maintaining high quality products can often lead to increased customer satisfaction and loyalty, resulting in higher sales and profits. It is important to strike a balance between cost-cutting measures and maintaining quality standards to maximize profit in the long run.

How often should cost and price for 300 units be reassessed?

Cost and price for 300 units should be regularly reassessed to ensure that they are still aligned with market conditions and business goals. This could mean reassessing every quarter, every year, or even more frequently depending on the industry and market volatility. It is important to stay proactive and make adjustments as needed to remain competitive and maximize profit.

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