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The magnitude of the energy difference is 1.889 eV.
A science communicator explains radioactive decay using a sample with a half-life of 8 years. If the initial mass is 200 grams, how much remains after 24 years?
Number of half-lives: $ 24 / 8 = 3 $.
Remaining mass = $ 200 \times (1/2)^3 = 200 \times 1/8 = 25 $ grams.
A seismologist analyzes wave velocities through layers of Earth, finding that P-wave speed increases linearly with depth. At 10 km depth, speed is 6 km/s; at 30 km, it’s 7.5 km/s. What is the speed at 22 km?
The rate of increase is $ (7.5 - 6) / (30 - 10) = 1.5 / 20 = 0.075 $ km/s per km.
From 10 km to 22 km is 12 km, so increase = $ 12 \times 0.075 = 0.9 $ km/s.
Speed at 22 km = $ 6 + 0.9 = 6.9 $ km/s.
A theoretical physicist models dark energy density as a constant force field with strength $ \rho = k/ r^2 $, where $ r $ is distance from a point source. If $ \rho = 64 $ units at $ r = 2 $, what is $ \rho $ at $ r = 4 $?
Since $ \rho = k / r^2 $, then $ k = \rho \cdot r^2 = 64 \cdot 2^2 = 64 \cdot 4 = 256 $.