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Home » الحل: نطبق خوارزمية إقليدس لإيجاد $\gcd(105, 165)$. - Silent Sales Machine

الحل: نطبق خوارزمية إقليدس لإيجاد $\gcd(105, 165)$. - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 t = \frac{\ln(2)}{0.08} \approx \frac{0.6931}{0.08} \approx 8.66 \text{ years}

📌 A conical tank with height 12 meters and base radius 4 meters is being drained at a rate of 3 m³/min. How fast is the water level decreasing when the water is 6 meters deep?

📌 V = \frac{1}{3}\pi r^2 h

📌 By similar triangles, \( \frac{r}{h} = \frac{4}{12} = \frac{1}{3} \), so \( r = \frac{h}{3} \).

📌 Substitute into volume:

📌 V = \frac{1}{3}\pi \left(\frac{h}{3}\right)^2 h = \frac{1}{3}\pi \frac{h^3}{9} = \frac{\pi}{27} h^3

📌 Differentiate with respect to time:

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\mathbf{M} egin{bmatrix} x \ y \end{bmatrix} = egin{bmatrix} x + 2y...
This implies that \(\mathbf{M}\) transforms \((x, y)\) into \((x + 2y,...
\mathbf{M} egin{bmatrix} x \ y \end{bmatrix} = egin{bmatrix} a & b \...
Equating components:
ax + by = x + 2y \quad \Rightarrow \quad a = 1, \; b = 2
cx + dy = 3x - y \quad \Rightarrow \quad c = 3, \; d = -1
Thus, the matrix \(\mathbf{M}\) is:
\mathbf{M} = egin{bmatrix} 1 & 2 \ 3 & -1 \end{bmatrix}
oxed{egin{bmatrix} 1 & 2 \ 3 & -1 \end{bmatrix}}
A science communicator is creating a vector visualization for a YouTub...

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