- 1000
- 2700
- 3200
- 4100
- 5300

Solution :

Which of the following is the best approximation of $\sqrt{1.5}(266)^{3/2}$?

**Choice 5 is the answer.**

With some simplification, we note that: \[\begin{eqnarray*}

\sqrt{1.5}(266)^{3/2}=\sqrt{\frac{3}{2}}\cdot(266)^{3/2} & = & 266\sqrt{\frac{3\cdot 266}{2}} \\

& = & 266\sqrt{399}\approx266\cdot20\approx 5300.

\end{eqnarray*}\]

Those of us familiar with Taylor series may try approximations with calculus, however this question teaches us that too much knowledge may be a little bit dangerous if misapplied. And that most of the time, there are simpler solutions than you'd think.

- 1000
- 2700
- 3200
- 4100
- 5300

Solution :

With some simplification, we note that: \[\begin{eqnarray*}

\sqrt{1.5}(266)^{3/2}=\sqrt{\frac{3}{2}}\cdot(266)^{3/2} & = & 266\sqrt{\frac{3\cdot 266}{2}} \\

& = & 266\sqrt{399}\approx266\cdot20\approx 5300.

\end{eqnarray*}\]

Those of us familiar with Taylor series may try approximations with calculus, however this question teaches us that too much knowledge may be a little bit dangerous if misapplied. And that most of the time, there are simpler solutions than you'd think.

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[; 1.5 * 250^3 \sim 1.5 * 16000000 = 24000000.;]

The square root of that is nearly $5000$, so the answer will be $5300$.

Hmm but you rounded up when you approximated 1.5*16000000 and then you rounded up again when you approximated 24000000 as 25000000 so I'm not entirely convinced.

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