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Home » 2: The Surprising Payoff You’re Avoiding If You’re Given Everything - Aurero

2: The Surprising Payoff You’re Avoiding If You’re Given Everything - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 A_{\text{original}} = \frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2

📌 After increasing each leg by 2 cm, the new legs are 10 cm and 17 cm. The new area is:

📌 A_{\text{new}} = \frac{1}{2} \times 10 \times 17 = 85 \text{ cm}^2

📌 - 60 = 25 \text{ cm}^2

📌 Thus, the area increases by:

📌 \boxed{25}

📌 Question:** Find \(\sin 315^\circ\).

📌 Since \(315^\circ = 360^\circ - 45^\circ\), we use the reference angle and the sine quadrant rule:

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\tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}
Rationalize the denominator:
\]Question: A weather balloon is modeled as a sphere with radius $ r $...
Solution: The volume of a sphere is given by $ V = \frac{4}{3} \pi r^3...
We are given that $ \frac{dV}{dt} = k \sin \theta $, and $ \theta = 60...
k \cdot \frac{\sqrt{3}}{2} = 4\pi r^2 \frac{dr}{dt}
Solving for $ \frac{dr}{dt} $:
\frac{dr}{dt} = \frac{k \sqrt{3}}{8\pi r^2}
The rate of volume increase is given directly as $ \frac{dV}{dt} = k \...
\frac{dV}{dt} = \frac{k\sqrt{3}}{2}

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