THE CUBOID CHALLENGE CHALLENGE CHALLENGE ACCEPTED Rough Figure
THE CUBOID CHALLENGE
CHALLENGE
CHALLENGE ACCEPTED!!! Rough Figure The side of the square sheet is 20 x 20 cm. Let the side of square to be cut be ‘x’. Now the new length of the box will be (20 -2 x). • x cannot be 0 or less than it as it would imply to not cutting any square, or in other words x > 0 • x cannot be more than or equal 10 as well. This is because we will end up cutting one whole side of the square. If x=0, (20 -2 x) = 20 -2(10) =0 • So the value of x should be 0 < x < 10. Now we are ready to make trials on this!!
TRIAL - 1 Here, x=1 i. e. side of square cut is 1. So, Volume = l. b. h = (20 -2 x)(x) = (20 -2 x) 2. (x) = (20 -2(1)) 2. (1) = 324 cm 3
TRIAL - 2 Here, x=2 i. e. side of square cut is 2. So, Volume = l. b. h = (20 -2 x)(x) = (20 -2 x) 2. (x) = (20 -2(2)) 2. (2) = 512 cm 3
TRIAL - 3 Here, x=3 i. e. side of square cut is 3. So, Volume = l. b. h = (20 -2 x)(x) = (20 -2 x) 2. (x) = (20 -2(3)) 2. (3) = 588 cm 3
TRIAL - 4 Here, x=4 i. e. side of square cut is 4. So, Volume = l. b. h = (20 -2 x)(x) = (20 -2 x) 2. (x) = (20 -2(4)) 2. (4) = 576 cm 3
TRIAL - 5 Here, x=5 i. e. side of square cut is 5. So, Volume = l. b. h = (20 -2 x)(x) = (20 -2 x) 2. (x) = (20 -2(5)) 2. (5) = 500 cm 3 TRIAL - 1
Let’s tabulate it!! ATTEMPT LENGTH BREADTH HEIGHT VOLUME TRIAL 1 18 cm 1 cm 324 cm 3 TRIAL 2 16 cm 2 cm 512 cm 3 TRIAL 3 14 cm 3 cm 588 cm 3 TRIAL 4 12 cm 4 cm 576 cm 3 TRIAL 5 10 cm 500 cm 3 From this we found that the maximum volume was attained when the square cut measured between 3 cm and 4 cm. Now let’s find more accurate answer by taking decimal places for the same.
Getting an accurate answer ATTEMPT LENGTH OF SQUARE CUT LENGTH BREADTH HEIGHT VOLUME TRIAL 1 3. 1 cm 13. 8 cm 3. 1 cm 590. 364 cm 3 TRIAL 2 3. 2 cm 13. 6 cm 3. 2 cm 591. 872 cm 3 TRIAL 3 3. 3 cm 13. 4 cm 3. 3 cm 592. 548 cm 3 TRIAL 4 3. 4 cm 13. 2 cm 3. 4 cm 592. 416 cm 3 TRIAL 5 3. 5 cm 13 cm 3. 5 cm 591. 500 cm 3 Here we see that we get largest volume when the measure of square cut is between 3. 3 to 3. 4 How about getting a step closer!!
One more step to get closer!! ATTEMPT LENGTH OF SQUARE CUT LENGTH BREADTH HEIGHT VOLUME TRIAL 1 3. 31 cm 13. 38 cm 3. 31 cm 592. 570764 cm 3 TRIAL 2 3. 32 cm 13. 36 cm 3. 32 cm 592. 585472 cm 3 TRIAL 3 3. 33 cm 13. 34 cm 3. 33 cm 592148 cm 3 TRIAL 4 3. 34 cm 13. 32 cm 3. 34 cm 592. 590816 cm 3 So there we go!! The maximum volume for a 20 cm by 20 cm box is 592148 cm 3. This is obtained when the square cut measures 3. 33 cm.
LET’S PROVE MATHEMATICALLY Let’s make the box (orange) by cutting squares (blue) of side ‘x’. To make the box of maximum volume, the value of ‘x’ should be optimum. So, by using maxima and minima concept, we can prove the optimum value of ‘x’ to get maximum volume. Let function f(x) = (20– 2 x). (x) be the volume of the box. To get maximum and minimum value of f(x), its derivative f’(x) should equal to zero. Therefore, f(x) = (20 -2 x)(x) = (20 -2 x)2. (x) = (400 + 4 x 2 – 80 x). (x) = (4 x 2 – 80 x + 400). (x) = (4 x 3 – 80 x 2 + 400 x) Now differentiating f(x), f’(x) = (3. 4. x 2 – 2. 80. x + 400)
For maxima and minima, f’(x) = 0 Therefore, f’(x) = => Dividing by 4, (3. 4. x 2 – 2. 80. x + 400) = 0 12 x 2 – 160 x + 400 = 0 3 x 2 – 40 x + 100 = 0 3 x 2 – 30 x – 10 x + 100 = 0 3 x(x– 10) – 10(x– 10) = 0 (3 x – 10)(x– 10) = 0 Therefore, By applying these values in f(x), we get, f(10) = (20 -2(10)). (10) =0 Hence, we mathematically proved that, the maximum volume of the box is 592 cm 3 when the value of the length of square (x) cut is or 3. 33 cm.
Trying some more Now that we found the maximum volume of 20 cm by 20 cm sheet, let’s try some more. Next let’s take 12 cm by 12 cm square sheet and 24 cm by 24 cm square sheet. Let’s proceed by repeating the previous steps. Let’s now tabulate these for the same. (i) For 12 cm by 12 cm square sheet, Volume = (12 -2 x). (x) Value of x (square cut) 0 < x < 6 (ii) For 24 cm by 24 cm square sheet, Volume = (24 -2 x). (x) Value of x (square cut) 0 < x < 12
For 12 cm by 12 cm square sheet ATTEMPT x l b h TRIAL 1 1 cm 100 cm 3 TRIAL 2 2 cm 8 cm 2 cm 128 cm 3 l = Length TRIAL 3 3 cm 6 cm 3 cm 108 cm 3 b = Breadth TRIAL 4 4 cm 64 cm 3 TRIAL 5 5 cm 2 cm 5 cm 20 cm 3 Here, x = Length of square cut h = Height The table at the top shows us the different volumes obtained by varying the value of ‘x’. The maximum volume is obtained when the square cut is between 2 cm and 3 cm. The two tables at the bottom show a much closer approximate of 2. As the trend of decreasing volume is observed, we conclude that maximum volume for a box of 12 cm by 12 cm is 128 cm 3 when size of square-cut is 2 cm. VOLUME ATTEMPT x l b h VOLUME TRIAL 1 2. 1 cm 7. 8 cm 2. 1 cm 127. 764 cm TRIAL 2 2. 2 cm 7. 6 cm 2. 2 cm 127. 072 cm TRIAL 3 2. 3 cm 7. 4 cm 2. 3 cm 125. 948 cm ATTEMPT x l b h VOLUME TRIAL 1 1. 8 cm 8. 4 cm 1. 8 cm 127. 008 cm TRIAL 2 1. 9 cm 8. 2 cm 1. 9 cm 127. 756 cm 3 3 3
For 24 cm by 24 cm square sheet ATTEMPT x l b h TRIAL 1 1 cm 22 cm 1 cm 484 cm 3 TRIAL 2 2 cm 20 cm 2 cm 800 cm 3 TRIAL 3 3 cm 18 cm 3 cm 972 cm 3 b = Breadth TRIAL 4 4 cm 16 cm 4 cm 1024 cm h = Height TRIAL 5 5 cm 14 cm 5 cm 980 cm ATTEMPT x l b h VOLUME The maximum volume is obtained when the square cut is between 4 cm and 5 cm. TRIAL 1 4. 1 cm 15. 8 cm 4. 1 cm 1023. 524 cm TRIAL 2 4. 2 cm 15. 6 cm 4. 2 cm 1022. 112 cm The two tables at the bottom show a much closer approximate of 4. TRIAL 3 4. 3 cm 15. 4 cm 4. 3 cm 1019. 788 cm As the trend of decreasing volume is observed, we conclude that maximum volume for a box of 24 cm by 24 cm is 1024 cm 3 when size of square-cut is 4 cm. ATTEMPT x l b h VOLUME TRIAL 1 3. 8 cm 16. 4 cm 3. 8 cm 1022. 048 cm 3 TRIAL 2 3. 9 cm 16. 2 cm 3. 9 cm 1023. 516 cm 3 Here, x = Length of square cut l = Length The table at the top shows us the different volumes obtained by varying the value of ‘x’. VOLUME 3 3 3
OBSERVATIONS SIZE OF PAPER SIZE OF SQUARE CUT MAXIMUM VOLUME 20 cm by 20 cm 3. 33 cm 592 cm 3 12 cm by 12 cm 2 cm 128 cm 3 24 cm by 24 cm 4 cm 1024 cm 3 Observing these results, we come up with a conclusion that maximum volume occurs when square cut out is one-sixth of the original square sheet. EXAMPLE: SIZE OF PAPER SIZE OF SQUARE CUT 45 7. 50 28 4. 66 32 5. 33 If we repeat the same process, then we will definitely get these results
CHALLENGE COMPLETED!!! THANK YOU!! BY: GR 090115 Grade 9
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