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Math Help - Short Calculus Word Problem

  1. #1
    Affinity
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    Smile Short Calculus Word Problem

    A closed rectangular box is to be constructed with one side 1 meter long. The material for the top of the box cost $20 per square meter, and the material for the sides and bottom costs $10 per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of $240 in materials. (Thanks in advance!)
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  2. #2
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    Quote Originally Posted by Affinity View Post
    A closed rectangular box is to be constructed with one side 1 meter long. The material for the top of the box cost $20 per square meter, and the material for the sides and bottom costs $10 per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of $240 in materials. (Thanks in advance!)
    hello,

    1. main condition (contains the value which will become extreme): V=w \cdot h \cdot 1

    2. additional conditions:
    A_{\text{top}}=w \cdot 1

    A_{\text{side and bottom}}=w \cdot 1+2 \cdot w \cdot h+2\cdot h\cdot 1

    costs:
    \text{costs}=20w + 10\cdot (w \cdot 1+2 \cdot w \cdot h+2\cdot h\cdot 1)=240 . Solve for h and plug this term into the equation of the main condition:

    h=\frac{24-3w}{2w+1}. You'll get the characteristic function of this problem:

    V(w)=\frac{24w-3w^2}{2w+1}

    Now calculate the derivative of V: V'(w)=\frac{-6w^2-6w+24}{(2w+1)^2}

    V has an extreme value (maximum or minimum) if V'(w) = 0. Calculate for w. (V'(w) = 0 if the numerator is zero and the denominator is not zero). You'll get two values for w. Only the positive value makes some sense.
    I've got w=\frac{-1+\sqrt{17}}{2}\approx 1.56\ m

    My result for h = 4.68 m

    EB
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  3. #3
    Affinity
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    Hi, I tried to redo the problem for myself, but for the denominator it came out as 2w+2 as opposed to your 2w+1, is there something I'm doing wrong? Thanks!
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  4. #4
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    Quote Originally Posted by Affinity View Post
    Hi, I tried to redo the problem for myself, but for the denominator it came out as 2w+2 as opposed to your 2w+1, is there something I'm doing wrong? Thanks!
    Hello,

    no, you have done the calculation perfectly.

    I've checked my calculations again and I got now: w = 2 m and h = 3 m

    (But as you kow now: I am the master of desaster if it comes to calculation, so if your results differ from these values - don't trust me!)

    EB
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  5. #5
    Affinity
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    Hi!

    I got the same answer as you did! Yay! Thank you very much for your help !
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