Carbon- 14 is an element which loses exactly half of its mass every 5730 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 741 grams. Write a function that models the mass of the carbon- 14 sample remaining t years since the initial measurement. M(t)=◻

Identify initial mass and decay factor: Identify the initial mass (a) and the decay factor (b). The initial mass of the carbon-14 sample is given as 741 grams. This is the value of 'a' in our exponential decay function. Since the mass of carbon-14 loses half of its mass every 5730 years, the decay factor 'b' is 1/2 (because every 5730 years, the mass is multiplied by 1/2).Determine decay rate per year: Determine the decay rate per year. The decay rate per year is not directly given, but we know that the mass halves every 5730 years. Therefore, the decay factor 'b' is raised to the power of t/5730, where t is the number of years since the initial measurement.Write exponential decay function: Write the exponential decay function. The general form of an exponential decay function is M(t) = a(b)^t. In this case, 'a' is the initial mass, and 'b' is the decay factor raised to the power of t/5730. Therefore, the function that models the mass of the carbon-14 sample remaining t years since the initial measurement is: M(t) = 741(1/2)^(t/5730)

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Carbon $-14$ is an element which loses exactly half of its mass every $5730$ 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 $741$ grams. $\newline$ Write a function that models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. $\newline$ $M(t)=\square$

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Q. Carbon

-14

is an element which loses exactly half of its mass every

5730

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

741

grams.

\newline

Write a function that models the mass of the carbon-

14

sample remaining

t

years since the initial measurement.

\newline

M(t)=\square

Identify initial mass and decay factor: Identify the initial mass ( $a$ ) and the decay factor ( $b$ ). $\newline$ The initial mass of the carbon $-14$ sample is given as $741$ grams. This is the value of ' $a$ ' in our exponential decay function. $\newline$ Since the mass of carbon $-14$ loses half of its mass every $5730$ years, the decay factor ' $b$ ' is $\frac{1}{2}$ (because every $5730$ years, the mass is multiplied by $\frac{1}{2}$ ).
Determine decay rate per year: Determine the decay rate per year. $\newline$ The decay rate per year is not directly given, but we know that the mass halves every $5730$ years. Therefore, the decay factor ' $b$ ' is raised to the power of $\frac{t}{5730}$ , where $t$ is the number of years since the initial measurement.
Write exponential decay function: Write the exponential decay function. $\newline$ The general form of an exponential decay function is $M(t) = a(b)^t$ . In this case, $'a'$ is the initial mass, and $'b'$ is the decay factor raised to the power of $\frac{t}{5730}$ . Therefore, the function that models the mass of the carbon $-14$ sample remaining $t$ years since the initial measurement is: $\newline$ $M(t) = 741\left(\frac{1}{2}\right)^{\frac{t}{5730}}$