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A cone-shaped paper drinking cup is to be made to hold $ 27 cm^3 $ of water. Find the height and radius of the cup that will use the smallest amount of paper.

height $\approx 3.722 \mathrm{cm}, \quad$ radius $\approx 2.632 \mathrm{cm}$

05:43

Wen Z.

02:08

Amrita B.

14:55

Linda H.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Derivatives

Differentiation

Volume

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Harvey Mudd College

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we have a question and this acorn shaped paper drinking cup is to be made to hold 26 27 centimeter cube. So this is the corn. Okay find the height and radius of the cup so that with the smallest amount of paper, if this is the height, This is the radius. So volume bill is one x 3 by our cue, one x 3 buyers Square H. This is the volume And volume is being given as 27 centimetre q. So 27 will be equal to one x 3. Fire square H. So let us say that will be equal 27-381 by by R squared. Okay now we have to maximize or minimize the total surface area total surface area. So here there will be no base because it is a cup so called surface area. So curve surface area there is a C will be equal to by uh L so here by R. L will be equal to at the square place are square on the route at a square plus r squared and that's all. Okay so here we could have found out at the square And so we are square. Just for the sake of simplicity. Square is 81 by buy it. And this is a clip again we can get sequel to by our into yeah At a square plus our Square 81 by by H and are here This is also our I will be nine by under route By H and two At the Squire by 81 by by H. Okay So if you could just take it inside it will be nine at the Squire. So at by bye. Bless 85. When bye by square at the square. Okay. So now we have to differentiate this and we know that if we have to maximize or minimize this function see? So let us take this only and we can do rest of them. So let us say this function is uh an ach equal to catch by pi Plus 81 by by a square. It's square. So to maximize minimize and for critical point of that that should be equal to zero. So one by pi. Uh huh. Plus. Okay let us write that at by pi Plus 81 by Pi Square at the rest of the par -2. So bless 81 by Pi Square -2 as opposed to the par -3. Okay, so they should be equal to zero which means one by pi place 162 But by a square excuse should be equal to zero. Okay uh This should be negative. So 162 by By Square. HQ should be equal to one by PI. Mhm Which means that you will be equal to 162 by Pi. So it will be equal to 162 by by raised to the power one x 3. So this is the value of it. No to get the value of our From here we can get the value of our to our square is eight. Even by pi at our square is 81. Bye. Buy into 162. My pie one x 3. So this is 81 by buy into one by pirates. The part one by 31 day 62 raised the part one by three. Yeah so this is 81 by pirates to the power To buy 3 162. Mhm 333 333. So here it will be yeah three In 2 3 days to the part one x 3. So this is 27 which is three cube by pirates with the power to buy three trillions department by three. So are square will be equal to uh previous to the power eight by three Divide by pirates to the power to buy three. So I will be equal to the square root of three years to the power 813. Pirates with power to buy three one by two. So our will be equal to three days to the power four by three divide by pirates to the power one by three. So this is three In 2, 3 days to the part one by three divided by pirates to the parliament by three. So we have our which is equal to three and 23 by pi raised to the power one by three And h equals two 162 by pirates to the power on by three. Mhm. Ach is 162 by five years to the power one by three. Uh So this is three days to the power. Uh huh. Well the and nine. So simply this will be equal to three years to power 33 to 1 by three Pirates Department by three. Okay, so actually equal to three and 2 three days to the power to buy three divide by Pirates Department by three. So these are dancers. Thank you.

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