You are planning to make an open rectangular box from a 33​-in.-by-65​-in. piece of cardboard by cutting congruent squares from the corners

Question

You are planning to make an open rectangular box from a 33​-in.-by-65​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this​ way, and what is its​ volume?

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Adalynn 3 months 2021-09-18T10:16:15+00:00 1 Answer 0 views 0

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    2021-09-18T10:18:14+00:00

    Answer:

    Volume = 6783.27inch³

    H = 6.951inch

    L = 51.098inch

    W = 19.098inch

    Step-by-step explanation:

    From the given information:

    Let h = side length and height of the box

    L = length of the box

    W = width of the box

    V = volume of the box

    We have that:

    L = 65-2h

    W = 33-2h

    V = L×W×h

    Therefore we have

    V = (65-2h)(33-2h)×h

    V = (2145-130h-66h+4h²)×h

    V = 2145h-196h²+4h³

    By differentiating V w.r.t h, we have

    V’ = 2145-392h+12h²

    V = 12h²-392h+2145

    Using Almighty formula we have

    h = 392+/-√392²-4(12)(2145)/2(12)

    h = 6.951 or 25.716

    Thus, we find L,W and V.

    We use the least value of h in order not to get a negative value of volume, length and width

    L = 65-2h = 65-2(6.951) = 51.098

    W = 33-2h = 33-2(6.951) = 19.098

    V = LWH = 51.098×19.098×6.951 = 6783.2696inches

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