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Thread: Mathematical Models (PreCalc)

  1. #1
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    Mathematical Models (PreCalc)

    These are a few review problems I was given for a upcoming test and I'm completely stumped how to do these.

    My main problem is the concept of "Expressing :blank: as a function of :blank:".

    You can answer the problems completely if you'd like ( learning for my self would benefit me more but I could always do other problems ) but I just need a very dumbed down explanation.

    Example 1)

    The demand for a certain commodity is given $\displaystyle D(x) = -50x + 800; $ that is, x units of the commodity well be demanded by consumers when the price is $\displaystyle p = D(x) $ dollars pre unit. Total consumer expenditure $\displaystyle E(x) $ is the amount of money consumers pay to buy x units of the commodity.

    A: Express consumer expenditure as a function of x, and sketch the graph of $\displaystyle E(x) $.

    B: Use the in part A to determine the level of production x at which consumer expenditure is largest. What price $\displaystyle p $ corresponds to maximum consumer expenditure?

    Example 2)

    A spherical cell of radius r has volume $\displaystyle V = 4/3 pi r$
    and surface are $\displaystyle S = 4 pi r^2$.

    Express V as a function of S. If S is doubled, what happens to V?


    Thanks for any help any one gives.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ARR0624 View Post
    These are a few review problems I was given for a upcoming test and I'm completely stumped how to do these.

    My main problem is the concept of "Expressing :blank: as a function of :blank:".

    You can answer the problems completely if you'd like ( learning for my self would benefit me more but I could always do other problems ) but I just need a very dumbed down explanation.

    Example 1)

    The demand for a certain commodity is given $\displaystyle D(x) = -50x + 800; $ that is, x units of the commodity well be demanded by consumers when the price is $\displaystyle p = D(x) $ dollars pre unit. Total consumer expenditure $\displaystyle E(x) $ is the amount of money consumers pay to buy x units of the commodity.

    A: Express consumer expenditure as a function of x, and sketch the graph of $\displaystyle E(x) $.
    since the price is $\displaystyle D(x)$, the cost for buying $\displaystyle x$ units will be $\displaystyle xD(x)$ ...do you see why?

    B: Use the in part A to determine the level of production x at which consumer expenditure is largest. What price $\displaystyle p $ corresponds to maximum consumer expenditure?
    your answer to the last question should give you the formula for a parabola. the x coordinate of the vertex of this parabola is the level of production x at which consumer expenditure is largest. the y coordinate of the vertex of this parabola is the price that corresponds to maximum consumer expenditure

    Example 2)

    A spherical cell of radius r has volume $\displaystyle V = 4/3 pi r$
    and surface are $\displaystyle S = 4 pi r^2$.

    Express V as a function of S. If S is doubled, what happens to V?


    Thanks for any help any one gives.
    $\displaystyle V = \frac 43 \pi r^3$

    $\displaystyle S = 4 \pi r^2$

    we will try to put S in V

    $\displaystyle V = \frac 43 \pi r^3 = \frac 13 \cdot {\color {red}4 \pi r^2} \cdot r$

    can you take it from here?
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  3. #3
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    Wow, yeah that helps so much. I just had one of those "Omg Why didn't I see that" moments.

    I really appreciate it.
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