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Carbon-14 is an element which loses
(1)/(10) of its mass every 871 years. The mass of a sample of carbon-14 can be modeled by a function,
M, which depends on its age,
t (in years).
We measure that the initial mass of a sample of carbon- 14 is 960 grams.
Write a function that models the mass of the carbon- 14 sample remaining
t years since the initial measurement.

M(t)=

Carbon $-14$ is an element which loses $\frac{1}{10}$ of its mass every $871$ years. The mass of a sample of carbon $-14$ can be modeled by a function, $M$ , which depends on its age, $t$ (in years). $\newline$ We measure that the initial mass of a sample of carbon $-14$ is $960$ grams. $\newline$ Write a function that models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. $\newline$ $M(t)=$

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Write exponential functions: word problems

Full solution

Q. Carbon

-14

is an element which loses

\frac{1}{10}

of its mass every

871

years. The mass of a sample of carbon

-14

can be modeled by a function,

M

, which depends on its age,

t

(in years).

\newline

We measure that the initial mass of a sample of carbon

-14

is

960

grams.

\newline

Write a function that models the mass of the carbon-

14

sample remaining

t

years since the initial measurement.

\newline

M(t)=

Identify Mass and Decay Rate: Identify the initial mass $a$ and the decay rate $r$ . The initial mass $a$ is given as $960$ grams. The decay rate $r$ is given as losing $\frac{1}{10}$ of its mass every $871$ years.
Determine Decay Factor: Determine the decay factor $b$ . Since the sample loses $\frac{1}{10}$ of its mass every $871$ years, it retains $\frac{9}{10}$ of its mass every $871$ years. Therefore, the decay factor $b$ is $\frac{9}{10}$ .
Write Exponential Decay Function: Write the exponential decay function. $\newline$ The function that models the mass of the carbon $-14$ sample remaining after $t$ years is of the form $M(t) = a(b)^t$ . $\newline$ Here, ' $a$ ' is the initial mass, ' $b$ ' is the decay factor, and ' $t$ ' is the time in years.
Adjust for Time Period: Adjust the decay factor for the time period. $\newline$ Since the decay happens every $871$ years, we need to adjust the exponent to reflect the passage of time in terms of $871$ -year periods. $\newline$ The adjusted decay function is $M(t) = a(b)^{(t/871)}$ .
Substitute Values: Substitute the values of $a$ and $b$ into the function. $\newline$ Substitute $960$ for $a$ and $\frac{9}{10}$ for $b$ into the function $M(t) = a(b)^{\frac{t}{871}}$ . $\newline$ $M(t) = 960\left(\frac{9}{10}\right)^{\frac{t}{871}}$ .

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