the half life of a certain radioactive material is 68 hours an initial amount of the material has mass of 641 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 4 hours. Round your answer to the nearest thousandth

[tex]\Rightarrow0.5=e^{k\times68}\\\Rightarrow\ \ln(.5) = 68k\\\Rightarrow\ k=\frac{ln(.5)}{68}\\\Rightarrow\ k=\frac{-0.693147181}{68}\\\Rightarrow\ k=-0.010193341[/tex]

The required equation will be :

[tex]A=641e^{-0.0101933t}[/tex]

For t=4 years

[tex]A=641e^{-0.0101933\times4}\\\Rightarrow\ A=641e^{-0.0407734}\\\Rightarrow\ A = 641(0.960046687)\\\Rightarrow\ A=615.39\ kg[/tex]

Radioactive material remain after 4 years = 615.39 kg

ap8997154 ap8997154 · Answer 2 · 2022-05-03T13:55:13+02:00

The amount of material left after a period of 4 hours is 615.398 kg.

What is the general equation of exponential decay?

The general equation of the exponential decay is given as,

[tex]A = Pe^{kt}\\[/tex]

where A is the amount left after a time period t, k is the rate of decay, and P is the initial amount.

As it is given that the half-life of the radioactive material is 68 hours, therefore, the value of half-life can be written as,

[tex]0.5 = e^{kt}\\\\0.5 = e^{k \times 68}\\\\ln(0.5)= k \times 68\\\\k = -0.01019[/tex]

Now, as the rate of decay is known the exponential function that models the decay of this material is

[tex]A=641 \times e^{-0.01019t}[/tex]

Further, the amount of material left after 4 hours can be written as,

[tex]\rm A=641 \times e^{-0.01019t}\\\\A=641 \times e^{-0.01019 \times 4}\\\\A = 615.398\ kg[/tex]

Hence, the amount of material left after a period of 4 hours is 615.398 kg.

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