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Home » \[ f'(x) = 12x^2 - 10x + 1 \] - Redraw

\[ f'(x) = 12x^2 - 10x + 1 \] - Redraw

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 \(x = \frac{3 \pm \sqrt{25}}{4}\).

📌 \(x = \frac{3 \pm 5}{4}\).

📌 Roots: \(x = 2\) and \(x = -\frac{1}{2}\).

📌 #### 2, \(-\frac{1}{2}\)Question: A weather radar detects a storm front forming a triangle with sides measuring 10 km, 13 km, and 15 km. What is the length of the shortest altitude?

📌 Solution: First, compute the area of the triangle using Heron's formula. The semi-perimeter $ s = \frac{10 + 13 + 15}{2} = 19 $ km. The area $ A = \sqrt{19(19-10)(19-13)(19-15)} = \sqrt{19 \cdot 9 \cdot 6 \cdot 4} = \sqrt{4104} = 64.07 $ km² (approximate). The shortest altitude corresponds to the longest side (15 km). Using $ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} $, the altitude $ h = \frac{2A}{15} = \frac{2 \cdot 64.07}{15} \approx 8.54 $ km. For exact value, simplify $ \sqrt{4104} = \sqrt{4 \cdot 1026} = 2\sqrt{1026} $, but numerical approximation suffices here. Thus, the shortest altitude is $ \boxed{8.54} $ km.

📌 Question: A renewable energy microgrid uses a hemispherical solar reflector with radius $ 2x $ and a cylindrical battery storage tank with radius $ x $ and height $ 4x $. What is the ratio of the hemisphere's volume to the cylinder's volume?

📌 Solution: The volume of a hemisphere is $ \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (2x)^3 = \frac{16}{3}\pi x^3 $. The cylinder's volume is $ \pi r^2 h = \pi x^2 \cdot 4x = 4\pi x^3 $. The ratio is $ \frac{\frac{16}{3}\pi x^3}{4\pi x^3} = \frac{16}{3} \div 4 = \frac{4}{3} $. Thus, the ratio is $ \boxed{\dfrac{4}{3}} $.

📌 Question: A meteorologist models a storm's boundary as a circle enclosing a 3 cm by 4 cm rectangular sensor array. What is the circumference of the circle?

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