Question

The amount of radioactive uranium changes with time. The table below shows the amount of radioactive uranium f(t) left after time t:

t(hours)00.51f(t) 1005025Which exponential function best represents the relationship between f(t) and t?A).f(t) = 100(0.25)tB).f(t) = 0.25(100)tC).f(t) = 100 (0.5)tD).f(t) = 0.25(50)tI know the answer is either A or C, but I don’t quite understand why. Can you please help explain?

Answer 1

We can get the exponential function that defines the relationship between Uranium left and the time

The general formula for the exponential function is

`f(t)=ab^t`

The scope of the question requires that we find the value of a and b in the function above

For the first data, when t=0, f(t)=100

`\begin{gathered} 100=a* b^0 \\ 100=a*1 \\ 100=a \end{gathered}`

Thus

`a=100`

The next step will be to find b.

from the third data, when t=1, f(t)=25

`\begin{gathered} 25=a* b^1 \\ \text{but we now know that a=100} \end{gathered}`

we will then solve for b

`\begin{gathered} 25=100* b \\ b=(25)/(100)=0.25 \\ b=0.25 \end{gathered}`

The final step will be to substitute the values of a and b into the formula

`f(t)=100(0.25)^t`

The final answer is option A