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📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 \[ a = -\frac{g}{2v_0^2\cos^2\theta}, \quad b = \tan\theta. \]

📌 Thus, the horizontal distance $x$ at which the projectile reaches its maximum height is

📌 \[ x = -\frac{\tan\theta}{2\left(-\frac{g}{2v_0^2\cos^2\theta}\right)} = \frac{v_0^2\sin\theta\cos\theta}{g}. \]

📌 Using the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \), we rewrite the expression as

📌 \[ x = \frac{v_0^2 \sin(2\theta)}{2g}. \]

📌 Therefore, the horizontal distance is

📌 \[\boxed{\frac{v_0^2 \sin(2\theta)}{2g}}.\]

📌 A deep-sea extremophile genomics researcher is studying the growth dynamics of microbial communities near hydrothermal vents. The population of a certain microbial species is modeled by the equation \( P(t) = P_0 e^{kt} \), where \( k \) is a growth constant. If the population triples in 5 hours, what is the value of \( k \)?

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