The half-life of a certain radioactive material is 75 days. An initial amount of the material has a mass of 381 kg. How do you write an exponential function that models the decay of this material and how much radioactive material remains after 15 days?

Question

Jace Anderson · Accepted Answer

undefined The exponential function that models the decay of the radioactive material is:

$$ N(t) = N_0 \cdot e^{-\frac{t}{T_{\frac{1}{2}}}} $$

Where:
- $ N(t) $ is the amount of radioactive material remaining after time $ t $.
- $ N_0 $ is the initial amount of radioactive material.
- $ T_{\frac{1}{2}} $ is the half-life of the radioactive material.
- $ e $ is the base of the natural logarithm.

Given that $ N_0 = 381 $ kg and $ T_{\frac{1}{2}} = 75 $ days, the exponential function becomes:

$$ N(t) = 381 \cdot e^{-\frac{t}{75}} $$

To find out how much radioactive material remains after 15 days, we substitute $ t = 15 $ into the equation:

$$ N(15) = 381 \cdot e^{-\frac{15}{75}} $$

$$ N(15) = 381 \cdot e^{-\frac{1}{5}} $$

$$ N(15) = 381 \cdot e^{-0.2} $$

$$ N(15) ≈ 381 \cdot 0.8187 $$

$$ N(15) ≈ 312.29 \text{ kg} $$

So, approximately 312.29 kg of radioactive material remains after 15 days.

Nathan Axel · Answer

Half life: #y=x*(1/2)^t# with #x# as the initial amount, #t# as #"time"/"half life"#, and #y# as the final amount. To find the answer, plug in the formula: #y=381*(1/2)^(15/75)=># #y=381*0.87055056329=># #y=331.679764616# The answer is approximately #331.68#