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Numerade Educator

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Problem 14 Medium Difficulty

A box with a square base and open top must have a volume of $ 32,000 cm^3 $. Find the dimensions of the box that minimize the amount of material used.

Answer

Side length of the square base must be 40 $\mathrm{cm}$
Height of the box must be 20 $\mathrm{cm}$

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Video Transcript

We know that if axe is the length of the sides and H is the height that we know the volume is ax squared times h because remember the faces square that's by excess square. It's the same value when we know the volume is 32,000. Therefore, we know H is the volume 32,000 divided by X squared. Okay, now that we have this, we know we can write out surface area X squared plus four x times h Will we just establish with H is hello again. This gives us ax squared plus 128,000 over axe. Okay, take the derivative. So this equal to zero we end up with Axe is 40. So now that we have, we know surface area double primes, This means the second derivative is too because 250 sex 1000 over X cubed. So what we know is that X equals 40 haas to be a minimum. Therefore, we know we're using the least amount of materials possible when we have 32,000 divided by 40 r X squared, which is equivalent to 20. So this means that the side length of the square base is 40 centimeters and the height is going to be 20 centimeters