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)=

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 1/10 of its mass every 871 years.Determine Decay Factor: Determine the decay factor (b). Since the sample loses 1/10 of its mass every 871 years, it retains 9/10 of its mass every 871 years. Therefore, the decay factor (b) is 9/10.Write Exponential Decay Function: Write the exponential decay function. The function that models the mass of the carbon-14 sample remaining after t years is of the form M(t) = a(b)^t. 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. 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. The adjusted decay function is M(t) = a(b)^(t/871).Substitute Values: Substitute the values of 'a' and 'b' into the function. Substitute 960 for 'a' and 9/10 for 'b' into the function M(t) = a(b)^(t/871). M(t) = 960(9/10)^(t/871).

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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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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

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 $M(t) = a(b)^{\frac{t}{871}}$ . $\newline$ $M(t) = 960\left(\frac{9}{10}\right)^{\frac{t}{871}}$ .