Question

Math

Posted 4 months ago

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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).
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.
$M(t)=$
$\square$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To model the decay of Carbon-14, we use the exponential decay formula, which is $M(t) = M_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}$, where $M_0$ is the initial mass and $T$ is the half-life of the substance

step 2

Given that the initial mass $M_0$ is 741 grams and the half-life $T$ is 5730 years, we substitute these values into the formula to get $M(t) = 741 \times \left(\frac{1}{2}\right)^{\frac{t}{5730}}$

step 3

Simplifying the expression, we have $M(t) = 741 \times 0.5^{\frac{t}{5730}}$. This is the function that models the remaining mass of the Carbon-14 sample after $t$ years

[1] Answer

$M(t) = 741 \times 0.5^{\frac{t}{5730}}$

Key Concept

Exponential Decay

Explanation

The function $M(t) = 741 \times 0.5^{\frac{t}{5730}}$ represents the exponential decay of Carbon-14, where the mass halves every 5730 years.

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