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Problem 142

AMC 10/12, early questions

Algebra Difficulty 3.4 Find the answer

The formula $N=8 \times 10^{8} \times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars.
The lowest income, in dollars, of the wealthiest $800$ individuals is at least:
$\textbf{(A)}\ 10^4\qquad \textbf{(B)}\ 10^6\qquad \textbf{(C)}\ 10^8\qquad \textbf{(D)}\ 10^{12} \qquad \textbf{(E)}\ 10^{16}$

Your answer

Official solution

Plug $800$ for $N$ because $800$ is the number of people who has at least $x$ dollars.
$800 = 8 \times 10^{8} \times x^{-3/2}$
$10^{-6} = x^{-3/2}$
In order to undo raising to the $-\frac{3}{2}$ power, raise both sides to the $-\frac{2}{3}$ power.
$(10^{-6})^{-2/3} = (x^{-3/2})^{-2/3}$
$x = 10^4 \text{ dollars}$
The answer is $\boxed{\textbf{(A)}}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.