xl bullies - All Square Golf

🔗 Related Articles You Might Like:

📰 Correct approach: The gear with 48 rotations/min makes a rotation every $ \frac{1}{48} $ minutes. The other every $ \frac{1}{72} $ minutes. They align when both complete integer numbers of rotations and the total time is the same. So $ t $ must satisfy $ t = 48 a = 72 b $ for integers $ a, b $. So $ t = \mathrm{LCM}(48, 72) $. 📰 $ \mathrm{GCD}(48, 72) = 24 $, so $ \mathrm{LCM}(48, 72) = \frac{48 \cdot 72}{24} = 48 \cdot 3 = 144 $. 📰 Thus, after $ \boxed{144} $ seconds, both gears complete an integer number of rotations (48×3 = 144, 72×2 = 144) and align again. But the question asks "after how many minutes?" So $ 144 / 60 = 2.4 $ minutes. But let's reframe: The time until alignment is the least $ t $ such that $ 48t $ and $ 72t $ are both multiples of 1 rotation — but since they rotate continuously, alignment occurs when the angular displacement is a common multiple of $ 360^\circ $. Angular speed: 48 rpm → $ 48 \times 360^\circ = 17280^\circ/\text{min} $. 72 rpm → $ 25920^\circ/\text{min} $. But better: rotation rate is $ 48 $ rotations per minute, each $ 360^\circ $, so relative motion repeats every $ \frac{360}{\mathrm{GCD}(48,72)} $ minutes? Standard and simpler: The time between alignments is $ \frac{360}{\mathrm{GCD}(48,72)} $ seconds? No — the relative rotation repeats when the difference in rotations is integer. The time until alignment is $ \frac{360}{\mathrm{GCD}(48,72)} $ minutes? No — correct formula: For two polygons rotating at $ a $ and $ b $ rpm, the alignment time in minutes is $ \frac{1}{\mathrm{GCD}(a,b)} \times \frac{1}{\text{some factor}} $? Actually, the number of rotations completed by both must align modulo full cycles. The time until both return to starting orientation is $ \mathrm{LCM}(T_1, T_2) $, where $ T_1 = \frac{1}{a}, T_2 = \frac{1}{b} $. LCM of fractions: $ \mathrm{LCM}\left(\frac{1}{a}, \frac{1}{b}\right) = \frac{1}{\mathrm{GCD}(a,b)} $? No — actually, $ \mathrm{LCM}(1/a, 1/b) = \frac{1}{\mathrm{GCD}(a,b)} $ only if $ a,b $ integers? Try: GCD(48,72)=24. The first gear completes a rotation every $ 1/48 $ min. The second $ 1/72 $ min. The LCM of the two periods is $ \mathrm{LCM}(1/48, 1/72) = \frac{1}{\mathrm{GCD}(48,72)} = \frac{1}{24} $ min? That can’t be — too small. Actually, the time until both complete an integer number of rotations is $ \mathrm{LCM}(48,72) $ in terms of number of rotations, and since they rotate simultaneously, the time is $ \frac{\mathrm{LCM}(48,72)}{ \text{LCM}(\text{cyclic steps}} ) $? No — correct: The time $ t $ satisfies $ 48t \in \mathbb{Z} $ and $ 72t \in \mathbb{Z} $? No — they complete full rotations, so $ t $ must be such that $ 48t $ and $ 72t $ are integers? Yes! Because each rotation takes $ 1/48 $ minutes, so after $ t $ minutes, number of rotations is $ 48t $, which must be integer for full rotation. But alignment occurs when both are back to start, which happens when $ 48t $ and $ 72t $ are both integers and the angular positions coincide — but since both rotate continuously, they realign whenever both have completed integer rotations — but the first time both have completed integer rotations is at $ t = \frac{1}{\mathrm{GCD}(48,72)} = \frac{1}{24} $ min? No: $ t $ must satisfy $ 48t = a $, $ 72t = b $, $ a,b \in \mathbb{Z} $. So $ t = \frac{a}{48} = \frac{b}{72} $, so $ \frac{a}{48} = \frac{b}{72} \Rightarrow 72a = 48b \Rightarrow 3a = 2b $. Smallest solution: $ a=2, b=3 $, so $ t = \frac{2}{48} = \frac{1}{24} $ minutes. So alignment occurs every $ \frac{1}{24} $ minutes? That is 15 seconds. But $ 48 \times \frac{1}{24} = 2 $ rotations, $ 72 \times \frac{1}{24} = 3 $ rotations — yes, both complete integer rotations. So alignment every $ \frac{1}{24} $ minutes. But the question asks after how many minutes — so the fundamental period is $ \frac{1}{24} $ minutes? But that seems too small. However, the problem likely intends the time until both return to identical position modulo full rotation, which is indeed $ \frac{1}{24} $ minutes? But let's check: after 0.04166... min (1/24), gear 1: 2 rotations, gear 2: 3 rotations — both complete full cycles — so aligned. But is there a larger time? Next: $ t = \frac{1}{24} \times n $, but the least is $ \frac{1}{24} $ minutes. But this contradicts intuition. Alternatively, sometimes alignment for gears with different teeth (but here it's same rotation rate translation) is defined as the time when both have spun to the same relative position — which for rotation alone, since they start aligned, happens when number of rotations differ by integer — yes, so $ t = \frac{k}{48} = \frac{m}{72} $, $ k,m \in \mathbb{Z} $, so $ \frac{k}{48} = \frac{m}{72} \Rightarrow 72k = 48m \Rightarrow 3k = 2m $, so smallest $ k=2, m=3 $, $ t = \frac{2}{48} = \frac{1}{24} $ minutes. So the time is $ \frac{1}{24} $ minutes. But the question likely expects minutes — and $ \frac{1}{24} $ is exact. However, let's reconsider the context: perhaps align means same angular position, which does happen every $ \frac{1}{24} $ min. But to match typical problem style, and given that the LCM of 48 and 72 is 144, and 1/144 is common — wait, no: LCM of the cycle lengths? The time until both return to start is LCM of the rotation periods in minutes: $ T_1 = 1/48 $, $ T_2 = 1/72 $. The LCM of two rational numbers $ a/b $ and $ c/d $ is $ \mathrm{LCM}(a,c)/\mathrm{GCD}(b,d) $? Standard formula: $ \mathrm{LCM}(1/48, 1/72) = \frac{ \mathrm{LCM}(1,1) }{ \mathrm{GCD}(48,72) } = \frac{1}{24} $. Yes. So $ t = \frac{1}{24} $ minutes. But the problem says after how many minutes, so the answer is $ \frac{1}{24} $. But this is unusual. Alternatively, perhaps 📰 A Cone Has A Base Radius Of 4 Cm And A Height Of 9 Cm What Is Its Volume 6704254 📰 Secondary Consumer Examples 9185011 📰 Youre Experimenting With These 17 Mold Illness Symptoms Before Its Too Late 8766555 📰 Papas Games Secrets Revealed Master The Madness In Minutes 6153864 📰 Kolton Stewart Movies And Tv Shows 7934815 📰 Wis Lottery 1166428 📰 Best Affordable Smart Tv 2280319 📰 You Wont Believe How Azure Blob Pricing Shaves Costs By 70Heres The Secret 8387949 📰 Hotels In Prague 3512828 📰 Francescas North 349611 📰 Timeline Of Assassins Creed Games 3151549 📰 Britt Lowers Naked Moment Blows Mind What Shes Hiding Is Unbelievable 8289912 📰 Pennsylvania Governor Residence Fire 9191872 📰 Kirkland Signature 4128937 📰 Final Fantasy Magic Unlocked Master These Boss Spells Today 8632050