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Let $\mathbb{Q}^+$ be the set of positive rational numbers with operation $a * b = \frac{ab}{2}$. What is the identity element $e$ such that $a * e = a$ for all $a \in \mathbb{Q}^+$?
For identity: $a * e = \frac{ae}{2} = a$, so $ae = 2a$, thus $e = 2$.
The production of ATP is faster in which type of respiration?
Anaerobic respiration (glycolysis + fermentation) produces ATP very quickly (e.g., in sprinting) but only 2 ATP per glucose. Aerobic respiration produces ~36 ATP but takes longer.
$d = 5.03894$ and $\overline{d}$ is the decimal expression for $d$ rounded to the nearest thousandth.
Compare:
Column A: The number of decimal places where $d$ and $\overline{d}$ differ
Column B: 4
When we round $d = 5.03894$ to the nearest thousandth, we get $\overline{d} = 5.039$ (since the fourth decimal place is 9, which is $\geq 5$, we round up).
Comparing digit by digit:
- Position 1 (units): both have 5 โ
- Position 2 (tenths): both have 0 โ
- Position 3 (hundredths): both have 3 โ
- Position 4 (thousandths): $d$ has 8, $\overline{d}$ has 9 โ
- Position 5 onwards: $d$ has 94, $\overline{d}$ has nothing
The positions where they differ are the thousandths place and beyond (positions 4, 5, 6...). If we count decimal places where values differ, we get 3 places (the thousandth, ten-thousandth, and hundred-thousandth positions).
Actually, re-reading: the question asks for decimal PLACES where they differ. Since $\overline{d}$ ends at thousandths, comparing through all decimal places in $d$: they differ at the 4th and 5th decimal positions. That's 2 places, which is less than 4.
Therefore Column B is greater.
1. Find Cube Side ($a$): The body diagonal of the cube equals the diameter of the sphere. $\sqrt{3}a = 2R \implies a = \frac{2R}{\sqrt{3}}$.
2. Find Cube Mass ($m$): Density $\rho = \frac{M}{\frac{4}{3}\pi R^3}$. Mass of cube $m = \rho \times a^3 = \left(\frac{3M}{4\pi R^3}\right) \times \left(\frac{8R^3}{3\sqrt{3}}\right) = \frac{2M}{\pi \sqrt{3}}$.
3. Moment of Inertia ($I$): For a cube about the center-face axis, $I = \frac{ma^2}{6}$.
$I = \frac{1}{6} \left( \frac{2M}{\pi \sqrt{3}} \right) \left( \frac{4R^2}{3} \right) = \frac{8MR^2}{18\pi \sqrt{3}} = \frac{4MR^2}{9\sqrt{3}\pi}$.
In France, system of government is:
France operates under a semi-presidential (mixed) system where both a President and a Prime Minister hold executive powers.
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