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p-tau and t-tau alzheimer's - RTA

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 V_s = \frac{4}{3} \pi r^3

📌 The volume \( V_c \) of a cylinder with radius \( r \) and height \( 3r \) is:

📌 V_c = \pi r^2 \cdot 3r = 3\pi r^3

📌 The ratio of the volume of the sphere to the volume of the cylinder is:

📌 \frac{V_s}{V_c} = \frac{\frac{4}{3} \pi r^3}{3\pi r^3} = \frac{4}{3} \cdot \frac{1}{3} = \frac{4}{9}

📌 Thus, the ratio is \(\boxed{\frac{4}{9}}\).

📌 Question:** A palynologist is analyzing pollen grains that are roughly spherical in shape with a radius of \( a \) micrometers. If the radius of a spore is \( \frac{a}{2} \) micrometers, what is the ratio of the surface area of a pollen grain to the surface area of a spore?

📌 The surface area \( A_p \) of a sphere with radius \( a \) is:

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