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Advanced Math / Nonlinear functions Difficulty: Medium

A sample of a certain isotope takes $29$ years to decay to half its original mass. The function $s (t) = 184 {(0.5)}^{\frac{t}{29}}$ gives the approximate mass of this isotope, in grams, that remains $t$ years after a $184$ -gram sample starts to decay. Which statement is the best interpretation of $s left parenthesis 87 right parenthesis equals 23$ in this context?

Approximately $23$ grams of the sample remains $87$ years after the sample starts to decay.

The mass of the sample has decreased by approximately $23$ grams $87$ years after the sample starts to decay.

The mass of the sample has decreased by approximately $87$ grams $23$ years after the sample starts to decay.

Approximately $87$ grams of the sample remains $23$ years after the sample starts to decay.

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Explanation

Choice A is correct. In the given function, $s (t)$ represents the approximate mass, in grams, of the sample that remains $t$ years after the sample starts to decay. It follows that the best interpretation of $s left parenthesis 87 right parenthesis equals 23$ is that approximately $23$ grams of the sample remains $87$ years after the sample starts to decay.

Choice B is incorrect. The mass of the sample has decreased by approximately $184 minus 23$ , or $161$ , grams, not $23$ grams, $87$ years after the sample starts to decay.

Choice C is incorrect. The mass of the sample has decreased by approximately $78$ grams, not $87$ grams, $23$ years after the sample starts to decay.

Choice D is incorrect. This would be the best interpretation of $s left parenthesis 23 right parenthesis equals 87$ , not $s left parenthesis 87 right parenthesis equals 23$ .