The mass of a radioactive sample, in grams, is modeled by the function M(t) = M_0 \left(\f… — SAT

The mass of a radioactive sample, in grams, is modeled by the function $M(t) = M_0 \left(\frac{1}{8}\right)^{\frac{t}{15}}$ , where $M_0$ is the initial mass of the sample and $t$ is the time, in minutes, since the sample started decaying. Which of the following functions correctly models the mass of the sample, in grams, in terms of $s$ seconds after the sample started decaying?

A
$M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{5}}$
B
$M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{2700}}$
$M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}$ Answer
D
$M(s) = M_0 \left(\frac{1}{2}\right)^{12s}$

Answer

The correct function modeling the mass in terms of seconds is

M(s) = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

To express the mass in terms of

s

seconds, we substitute

t = \frac{s}{60}

into the original decay function because the time in minutes is equal to the number of seconds divided by 60. This gives

M(s) = M_0 \left(\frac{1}{8}\right)^{\frac{s}{60 \times 15}} = M_0 \left(\frac{1}{8}\right)^{\frac{s}{900}}

. Next, since the options have a base of

\frac{1}{2}

, we rewrite the base

\frac{1}{8}

\left(\frac{1}{2}\right)^3

. Applying the power of a power rule

(x^a)^b = x^{ab}

, we multiply the exponent

\frac{s}{900}

by 3, which yields

M(s) = M_0 \left(\frac{1}{2}\right)^{3 \times \frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

Step-by-Step Solution

Relate the time variables

t

(in minutes) and

s

(in seconds).

Since 1 minute is equivalent to 60 seconds, the relationship is

t = \frac{s}{60}

This substitution allows the function to take time input in seconds instead of minutes.

Substitute

t = \frac{s}{60}

into the exponent of the original model.

M(s) = M_0 \left(\frac{1}{8}\right)^{\frac{s/60}{15}} = M_0 \left(\frac{1}{8}\right)^{\frac{s}{900}}

This updates the function variable to seconds and simplifies the fractional exponent.

Rewrite the base

\frac{1}{8}

as a power of

\frac{1}{2}

and apply exponent rules.

Since

\frac{1}{8} = \left(\frac{1}{2}\right)^3

, we write

M(s) = M_0 \left(\left(\frac{1}{2}\right)^3\right)^{\frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{3 \times \frac{s}{900}} = M_0 \left(\frac{1}{2}\right)^{\frac{s}{300}}

Applying the power of a power rule,

(a^m)^n = a^{mn}

, simplifies the function to its final form with a base of

\frac{1}{2}

Key Concept

Applying exponent rules to manipulate exponential bases and performing variable substitutions in contextual models.

Estimated Time:2m 0s

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