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$Carbon-14 is a radioactive isotope used to determine the age of samples of organic matter. The amount of carbon-14 in a sample decreases once the organism is no longer alive. The half-life of carbon-14 is approximately 5730 years. This means that every 5730 years, the amount of carbon-14 in an organism that is no longer living will be halved. Which of the following functions, C, models the fraction of carbon-14 remaining in a sample after t years? Choose 1 answer: C(t)=2^((t)/( 5730)) (B) C(t)=((1)/(2))^((t)/( 5730)) (C) C(t)=((1)/(2))^("втза ") (D) C(t)=2^(5730 t)$

Carbon $-14$ is a radioactive isotope used to determine the age of samples of organic matter. The amount of carbon $-14$ in a sample decreases once the organism is no longer alive. The half-life of carbon $-14$ is approximately $5730$ years. This means that every $5730$ years, the amount of carbon $-14$ in an organism that is no longer living will be halved. Which of the following functions, $C$ , models the fraction of carbon $-14$ remaining in a sample after $t$ years? $\newline$ Choose $1$ answer: $\newline$ (A) $C(t)=2^{\frac{t}{5730}}$ $\newline$ (B) $C(t)=\left(\frac{1}{2}\right)^{\frac{t}{5730}}$ $\newline$ (C) $C(t)=\left(\frac{1}{2}\right)^{5730 t}$ $\newline$ (D) $C(t)=2^{5730 t}$

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Math Problems

Grade 7

Evaluate two-variable equations: word problems

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Q. Carbon

-14

is a radioactive isotope used to determine the age of samples of organic matter. The amount of carbon

-14

in a sample decreases once the organism is no longer alive. The half-life of carbon

-14

is approximately

5730

years. This means that every

5730

years, the amount of carbon

-14

in an organism that is no longer living will be halved. Which of the following functions,

C

, models the fraction of carbon

-14

remaining in a sample after

t

years?

\newline

Choose

1

answer:

\newline

(A)

C(t)=2^{\frac{t}{5730}}

\newline

(B)

C(t)=\left(\frac{1}{2}\right)^{\frac{t}{5730}}

\newline

(C)

C(t)=\left(\frac{1}{2}\right)^{5730 t}

\newline

(D)

C(t)=2^{5730 t}

Half-life function representation: Since the half-life of carbon $-14$ is $5730$ years, the amount of carbon $-14$ is halved every $5730$ years. This means we need a function that represents halving the amount over time.
Exponential decay factor: The function should show that for every $5730$ years that pass, the exponent in the function should increase by $1$ , because the amount of carbon- $14$ is halved (multiplied by $1/2$ ) each time.
Analysis of Option (A): Option (A) $C(t)=2^{\left(\frac{t}{5730}\right)}$ suggests that the amount of carbon $-14$ doubles every $5730$ years, which is incorrect.
Analysis of Option (B): Option (B) $C(t)=\left(\frac{1}{2}\right)^{\frac{t}{5730}}$ correctly shows that the amount of carbon $-14$ is halved every $5730$ years, because raising $\left(\frac{1}{2}\right)$ to the power of $\frac{t}{5730}$ increases the exponent by $1$ for each $5730$ years, resulting in halving the amount.
Analysis of Option (C): Option (C) $C(t)=\left(\frac{1}{2}\right)^{\text{＂втза ＂}}$ contains a typo or nonsensical input in the exponent and does not make mathematical sense.