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@yuki2go дал ЭЛЕГАНТНОЕ аналитическое решение, которое правда не дает индивидуальных значений a, b и c. Чего, впрочем и не требовалось. Для их нахождения потребовались бы дополнительные шаги. Позволю объяснить решение по шагам, надеюсь будет полезно школьникам, подобным мне. 1. (abc)² = 100 * 200 * 300 = 6000000 Это следует из перемножения всех трех уравнений: (ab)(bc)(ca) = 100 * 200 * 300 2. abc = ±1000√6 Это квадратный корень из 6000000. Знак ± появляется, так как квадратный корень может быть положительным или отрицательным. 3. a + b + c = abc/bc + abc/ac + abc/ab Это ключевой шаг. Мы выражаем a, b и c через их произведение и попарные произведения. 4. = abc(1/200 + 1/300 + 1/100) Здесь мы заменяем bc, ac и ab их значениями из исходных уравнений. 5. = abc/100(1/2 + 1/3 + 1) Упрощаем выражение, вынося общий множитель. 6. = ±(55√6)/3 Подставляем значение abc = ±1000√6 и упрощаем: ±1000√6/100 * (1/2 + 1/3 + 1) = ±10√6 * (11/6) = ±(55√6)/3
It's a nice problem, but I guess you could have solved it faster if you had first isolated b , this way: b = 100/a; b = 200/c; c = 2a. I found a = 5 sqrt 6; c = 10 sqrt 6; b = (10/3) sqrt 6; a+b+c = (55/3) sqrt 6; but I did not have to deal with big numbers, like root of 6050. I wonder if my solution has something wrong, but I think it's easier. Anyway, thanks for the problem.
AB=100 B=100/A so from BC=200 we have 100C/A=200 ie C=2A Then from CA=300 we have 2A^2=300 ie A=sqrt(150) We can deduce that: C=2sqrt(150) since C=2A and B=200/(2sqrt(150))=100sqrt(150)/150
After 3:14 (the value of a^2, b^2 and c^2), you can continue by : ab > 0, bc > 0 and ac > 0 so a, b and c are positives. And now you know the value of a, b and c i.e the value of a + b + c. It's faster. Your way after 3:14 is beautiful, I appreciate your idea.
I started with, [ab = 100, bc = 200], [200 = 2 * 100], [bc = 2ab], cancel out the b's so that [c = 2a], then plug it into ca = 300 so that 2a^2 = 300 and solve for a.
Исходные уравнения: 1. a *b = 100 2. b *c = 200 3. c *a = 300 Шаг 1: Найдем b через a b = 100/a Шаг 2: Подставим b в уравнение (2) 100/a *c = 200 100c = 200a c = 2a Шаг 3: Подставим c = 2a в уравнение (3) c *a = 300 (2a) *a = 300 2a^2 = 300 a^2 = 150 a = √150= 5√6 Шаг 4: Найдем b через a b = 100/a = 100/5√6= 20√6 Шаг 5: Найдем c через a c = 2a = 2 *5√6= 10√6 Итак, решение: a = 5√6 b = 20√6 c = 10√6
(bc)/(ab) = 2 => c = 2a ac = 300 = 2a² => a = ± √150 => c = ± 2√150 ab = 100 => b = ± 100/√150 a + b + c = ± (3√150 + 100/√150) a + b + c = ± (550/√150) a + b + c = ± [550/(5√6)] a + b + c = ± (110/√6) a + b + c = ± (110√6/6) *a + b + c = ± (55√6/3)*
ab = 100 bc = 200 ca = 300 =========== Metode I ab*bc*ca = 100*200*300 = 100*6*10000 (abc)² = 100*6*10000 abc = ±10*√6*100 = ±1000√6 a = ab*ca/abc = 100*300/(±1000√6) a = ±30/√6 b = ab*bc/abc = 100*200/(±1000√6) b = ±20/√6 c = bc*ca/abc = 200*300/(±1000√6) c = ±60/√6 a + b + c = ±(30+20+60)/√6 a + b + c = ±110/√6 ========== Methode 2 ab/bc = a/c = 100/200 = 1/2 a/c = 1/2 --> c = 2a bc/ca = b/a = 200/300 = 2/3 b/a = 2/3 --> b = 2a/3 ca = 300 (2a)a = 300 2a² = 300 a² = 150 a = ±5√6 a + b + c = a + 2a/3 + 2a a + b + c = 11a/3 a + b + c = 11(±5√6)/3 a + b + c = ±(55√6)/3 a + b + c = ±110/√6 ========= Method 3 As video ========= • • • Do we other method? Comment below 🙏🏻🙏🏻🙏🏻
Your methods give the right answer (method 2 seems simplest), but the convention seems to be to put the sqrts upstairs so (55/3)*sqrt(6) is preferable to 110/sqrt(6)
