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Home » n_1 \sin(\theta_1) = n_2 \sin(\theta_2) - Get link 4share

n_1 \sin(\theta_1) = n_2 \sin(\theta_2) - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solve each equation:

📌 \(b - 3 = 7 \Rightarrow b = 10\)

📌 \(b - 3 = -7 \Rightarrow b = -4\)

📌 The sum of all solutions is:

📌 + (-4) = 6

📌 Thus, the sum of all values of \(b\) is \(\boxed{6}\).Question: A cartographer is designing a grid-based map system where two roads intersect at a point, represented by positive integers \(a\) and \(b\), such that the total length of the combined road segments is 1000 units. If the greatest common divisor of \(a\) and \(b\) determines the largest uniform segment size that fits both roads exactly, what is the largest possible value of \(\gcd(a, b)\)?

📌 Solution: Let \(d = \gcd(a, b)\). Then we can write \(a = d \cdot m\) and \(b = d \cdot n\), where \(m\) and \(n\) are coprime positive integers. The total road length is \(a + b = d(m + n) = 1000\). So \(d\) must divide 1000. To maximize \(d\), we minimize \(m + n\), subject to \(m\) and \(n\) being coprime positive integers. The smallest possible value of \(m + n\) is 2, which occurs when \(m = n = 1\), and they are coprime. This gives \(d = \frac{1000}{2} = 500\). Since \(m = 1\) and \(n = 1\) are coprime, this is valid. Therefore, the largest possible value of \(\gcd(a, b)\) is \(\boxed{500}\).

📌 Question: A medical researcher is analyzing immune response cycles in mice, where one immune marker peaks every 18 days and another every 30 days. If both markers peak today, after how many days will they next peak simultaneously, assuming the pattern repeats every least common multiple of their cycles?

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