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Home » \[ (x - 1)(x - 3) = 0 \] - Deep Underground Poetry

\[ (x - 1)(x - 3) = 0 \] - Deep Underground Poetry

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Speed each hour: Hour 1: 18, Hour 2: 30, Hour 3: 42, Hour 4: 54, Hour 5: 66 m/s.

📌 Distance = sum = 18 + 30 + 42 + 54 + 66.

📌 + 30 = 48; 48 + 42 = 90; 90 + 54 = 144; 144 + 66 = <<144+66=210>>210 meters.

📌 #### 210Question: In spherical coordinates $(\rho, \theta, \phi)$, a geographer models a region where the angle $\phi$ satisfies $\phi = \frac{\pi}{4}$. What shape does this equation describe?

📌 Solution: In spherical coordinates, $\phi$ is the polar angle measured from the positive $z$-axis. When $\phi$ is constant, the set of points forms a cone with its apex at the origin, opening symmetrically around the $z$-axis. For $\phi = \frac{\pi}{4}$, the cone makes a $45^\circ$ angle with the $z$-axis. Thus, the shape is a cone.

📌 \boxed{\text{Cone}}

📌 Question: A synthetic biology researcher models microbial growth directions in permafrost using unit vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$. If $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{w} = \mathbf{w} \cdot \mathbf{u} = \frac{1}{2}$, find the maximum value of $|\mathbf{u} + \mathbf{v} + \mathbf{w}|$.

📌 Solution: Let $\theta$ be the angle between any pair of vectors. Since $\mathbf{u} \cdot \mathbf{v} = \cos\theta = \frac{1}{2}$, $\theta = 60^\circ$. The magnitude squared is $|\mathbf{u} + \mathbf{v} + \mathbf{w}|^2 = 3 + 2(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}) = 3 + 2(3 \cdot \frac{1}{2}) = 3 + 3 = 6$. Thus, the maximum value is $\sqrt{6}$.

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