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Problem 35 Hard Difficulty

The top and bottom margins of a poster are each $ 6 cm $ and the side margins are each $ 4 cm $. If the area of printed material on the poster is fixed at $ 384 cm^2 $, find the dimensions of the poster with the smallest area.

Answer

24 $\mathrm{cm}$ by 36 $\mathrm{cm}$

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Video Transcript

we're told to top and bottom margins of the poster are each six centimeters, and the side margins are each four centimeters. The area of the printed material on the poster is fixed at 384 square seven years. We're asked to find the dimensions of the poster, the smallest area, so we'll let X and Y be the width and height to the printed area. And why is the height of the printed area? And so we have that X Times Y is the fixed area, 384 square centimeters. Therefore, as a function of X, why is equal to 384 over X, he told the total height of the poster, including 600 year margin at the top and bottom. That's why, plus 12 the total whip, including the forcing your margins X plus eight. So the area of the total poster A is X plus eight times Y plus 12. We can write why, as a function of X by substituting so yeah, A as a function of X is X plus eight times 384 over X plus 12. Boiling we get is 12 x Plus 3072. So over X plus 480 show Mhm. Now, to minimize the area of the poster will find the derivative a prime of X. This is 12 minus 3072 over X squared. Find the critical values beset the sequel to zero. So we have X squared equals Yeah, 3072 over 12. Yes. So x where is equal to 256 and therefore X, which is a positive quantity, is in fact, 16 in centimeters. Yeah. Now, the second derivative, a double prime of X is positive two times 3 72 over X cubed, which we know is going to be greater than zero for positive X. Therefore, it follows that it's greater than zero for the value of X. We just found X equals 16 and it follows that a has a relative minimum at X equals 16. Of course, it's the only minimum. So this is the absolute minimum as well. I wasn't shit. Mhm. Now the corresponding why value is 384 over X, which is 384 over 16. You should have seen wish this is 24. Therefore, the total poster width is X Plus eight, which is 16 plus eight or 24 in a total poster height is Y plus 12, which is 24 plus 12 or 36. Yeah, so it's 24 centimeters by 36 centimeters, thanks.