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Math Help - Need help with a confusing word problem

  1. #1
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    Need help with a confusing word problem

    A large pizza franchise delivers large pizzas for $16. The equation y=-3x^2+72x+480 describes the profit, y, based on the increase, x, in the price of their large pizzas. Determine the range of prices, to the nearest cent, for which the franchise could sell each pizza and still make a profit of at least $640.
    Could someone teach me how to solve this?

    PS: Knowing why your answer is correct will help me more than just knowing the answer, so please explain why your answer is correct.
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  2. #2
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    Hello, mathdonkey!

    A large pizza franchise delivers large pizzas for $16. The equation: . y\:=\:-3x^2+72x+480
    describes the profit, y, based on the increase, x, in the price of their large pizzas.

    Determine the range of prices, to the nearest cent, for which the franchise
    could sell each pizza and still make a profit of at least $640.
    We want: . -3x^2 + 72x + 480 \:\geq \:640 \quad\Rightarrow\quad -3x^2 + 72x - 160 \:\geq \:0

    We want to know when: . y \:=\:-3x^2+72x - 160 .is above the x-axis.

    Since the graph is a down-opening parabola, the curve is above the x-axis
    . . between its x-intercepts.


    For the x-intercepts, we have: . -3x^2+72x-160 \:=\:0

    Quadratic Formula: . x \;=\;\frac{72 \pm\sqrt{72^2 - 4(-3)(-160)}}{2(-3)} \;=\;\frac{36 \pm 4\sqrt{51}} {3}

    So we have: . x \;=\;\begin{array}{ccc}21.5219... &\approx & 21.52 \\ 2.4780... &\approx & 2.48 \end{array}


    The franchise can raise the price per pizza by $2.48 or by $21.52
    . . or anything in between, and still make at least $640.


    The price can range from \boxed{\$18.48\text{ to }\$37.52} per pizza.

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  3. #3
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    Thanks for the answer Soroban!

    I have a few questions though. Is that the only method of solving this and why should I use that method to solve this? What about the question requires that method? Strange questions! But I think they will help me better understand the thought process for answering this question.
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  4. #4
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    Hello,
    Quote Originally Posted by mathdonkey View Post
    Thanks for the answer Soroban!

    I have a few questions though. Is that the only method of solving this and why should I use that method to solve this? What about the question requires that method? Strange questions! But I think they will help me better understand the thought process for answering this question.
    In any situation, you will have to find x in :
    -3x^2+72x+480 \ge 640.
    This is the text transcription, this is why you can't avoid it.

    I can give you a slightly different approach (and longer !)... using the method of completing the square.

    -3x^2+72x-160 \ge 0

    I will multiply the inequality by -3, to get something a perfect square with x. Don't forget to change \ge into \le because we multiplied by a negative number.


    9x^2-216x+480 \le 0

    (3x)^2-2 \cdot (3x) \cdot {\color{red}36}+480 \le 0

    \overbrace{(3x)^2-2 \cdot (3x) \cdot {\color{red}36}+{\color{red}36}^2}^{(3x-36)^2}-\underbrace{{\color{red}36}^2+480}_{=-816} \le 0

    (3x-36)^2-816 \le 0

    (3x-36)^2 \le 816

    --> - \underbrace{\sqrt{816}}_{4 \sqrt{51}} \le 3x-36 \le \sqrt{816}

    36-4 \sqrt{51} \le 3x \le 36+4\sqrt{51}

    \boxed{\frac{36-4 \sqrt{51}}{3} \le x \le \frac{36+4 \sqrt{51}}{3}}

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  5. #5
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    Another strange question from mathdonkey: Why is the equation y=-3x^2+72x+480 used to describe the profit? What does each part of the equation mean in the context of the question? I have a hard time understanding math when I don't know the importance of these things.
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  6. #6
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    One of the things important to understand is the "Definition".

    The profit simply is defined to follow the structure provided. It is just a model. There are many other possible models. For some reason, perhaps because some member of the Pizza Board of Directors was helping his grandson with his analytic geometry last night, this is the model that was selected for profit for this enterprise. It doesn't have to mean anything else and may very well not mean anything else.

    All models have weaknesses. Many models are very convenient. In this case, the otherwise general quadratic with negative initial coefficient leads to a nice parabola when plotted on appropriate coordinate axes. We know things about parabolas. We can find minimum and maximum values rather easily. We can find zeros and symmetries rather easily. There very fact that it is convenient may have contributed to the model's selection.

    Most likely, I think, the model was selected because you are working in a section of your textbook concerned with quadratic equations. How's that for a reason?

    I have a hard time understanding math when I don't know the importance of these things.
    You may wish to let go of this. Do you REALLY know how important '2' is? What is '2', exactly? Is '2' more or less important than '3'? Are you sure? Can you prove it? I suspect you are quite comfortable with '2' and with '3', even though you really do not know exactly how important they are.

    Please do not find reasons why you do not understand unless you will be able to fix the reasons when you find them. Just to identify the impediment is not a useful practice.

    My views. I welcome others'.
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  7. #7
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    I appreciate you taking the time to respond TKHunny, but as someone who has trouble with math your response came off very patronizing and discouraging.

    I guess I learn differently than most people, because when given the extra attention to help me answer these questions I tend to thrive better than people who don't need help. But that's just me. I can always just answer the question, but what am I learning then? How to manipulate numbers so I can find answers to textbook questions? That's not very satisfying...

    I'm sure there are other people who are as curious as me, maybe they don't have the courage to ask their silly questions though?

    Again thank you for your response. I have a new queston though which may help me better understand this; what information did the member of the Pizza Board of Directors need to create the quadratic equation "y=-3x^2+72x+480"? I'm not sure if that makes sense, or if it can even be answered, but it seems like a good question to ask. If I understand how the equation was written I'll have that much more to work with when I try to solve it, right?
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  8. #8
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    Hi again

    Again thank you for your response. I have a new queston though which may help me better understand this; what information did the member of the Pizza Board of Directors need to create the quadratic equation "y=-3x^2+72x+480"? I'm not sure if that makes sense, or if it can even be answered, but it seems like a good question to ask. If I understand how the equation was written I'll have that much more to work with when I try to solve it, right?
    I think this is good curiosity

    There are two possibilities :
    - the author invented this problem, just to make you use what you have learnt. No matter what, the most important for him was to find a situation that could fit in

    - this is taken from reality. They observe or foresee values of the benefit if "the price is...." or "increase price of...", etc...
    There are methods that let you forecast expected values, based on these observations. "Linear Regression" can help you in it, it's part of numerical analysis.
    Um...well, I'm not specialized in it so what I've said is only thoughts, I'm sure other people may help you more for this
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