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Home » Contradiction. Hence, no such vector $\mathbfw$ exists. - Crankk.io

Contradiction. Hence, no such vector $\mathbfw$ exists. - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 5Question: A 5 cm by 8 cm rectangle is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of $\pi$.

📌 Solution: The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem, the diagonal $d$ is $\sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}$. Therefore, the radius of the circle is $\frac{\sqrt{89}}{2}$. The circumference is given by $2\pi r = 2\pi \cdot \frac{\sqrt{89}}{2} = \pi\sqrt{89}$.

📌 \boxed{\pi\sqrt{89}}

📌 Question: The radius of a sphere is $2x$ units and the radius of a hemisphere is $3x$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere?

📌 Solution: The volume of a sphere is $\frac{4}{3}\pi (2x)^3 = \frac{4}{3}\pi (8x^3) = \frac{32}{3}\pi x^3$. The volume of a hemisphere is $\frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi (27x^3) = 18\pi x^3$. The ratio of the volumes is:

📌 \frac{\frac{32}{3}\pi x^3}{18\pi x^3} = \frac{32}{3 \cdot 18} = \frac{32}{54} = \frac{16}{27}

📌 \boxed{\frac{16}{27}}

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