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Problem 15 Medium Difficulty

If $ 1200 cm^2 $ of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer

4000 $\mathrm{cm}^{3}$

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Video Transcript

Okay. We know the amount of material 1200 can be set equal to X squared. Remember, the boxes has an open top was for X y, and then veer Volume is gonna be X squared Y. So now, writing this first part in terms of why Rise 1200 minus x squared over four ax, which simplifies to 300 over axe minus X over four. Substituting into the equation, we have V is X squared times 300 over. Axe minus X, divided by four. Okay, now that we've got that we know we can differentiate the air volume to get 300 minus three x squared, divided by four. Set this equal to zero. So when we end up with X is a square to 400 which is 20 centimeters. Therefore, we know plugging this end v of 20 is gonna be 300 times 20 minus 20 cubed over four, which is for 1000 centimeters cute for volume.