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Learn Spanish Fast—Click Here to Transform Your Language Skills Today! - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 A ladder 10 meters long leans against a wall, forming a 75° angle with the ground. How high up the wall does the ladder reach? (Round to nearest tenth)

📌 Height = \( 10 \cdot \sin(75^\circ) \).

📌 \( \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \)

📌 = \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \left( \frac{\sqrt{3} + 1}{2} \right) = \frac{\sqrt{2}(\sqrt{3} + 1)}{4} \approx \frac{1.414 \cdot 2.732}{4} \approx \frac{3.863}{4} \approx 0.9658 \).

📌 So height ≈ \( 10 \times 0.9658 = 9.658 \), rounds to 9.7 meters.

📌 #### 9.7Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is 2025, representing the total number of seismic tremors recorded over two consecutive years?

📌 Solution: Let $a + b = 2025$. The greatest common divisor $\gcd(a, b)$ must divide $a + b = 2025$. The maximum possible value of $\gcd(a, b)$ occurs when $a$ and $b$ are both multiples of the largest divisor of 2025. Since $2025 = 45^2 = 3^4 \cdot 5^2$, its largest proper divisor is $2025 / 3 = 675$. Thus, $\gcd(a, b) = 675$ when $a = 675$ and $b = 1350$.

📌 \boxed{675}

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