Question

The amount of a radioactive material changes with time. The table below shows the amount of radioactive material f(t) left after time t: t(hours) 0 1 2 f(t) 180 90 45 Which exponential function best represents the relationship between f(t) and t?

Answer 1

Wow, that question leaves the table unformatted and doesn't list the exponential function.

t   f(t)
0 180
1 90
2 45

Go to this link:
http://www.1728.org/halflif2.htm

Lambda (Greek letter λ) = ln (2) / half-life
Lambda = .693147 / half-life
From the table, it seems that the half-life = 1 hour.
Lambda = .693147 / 1
Lambda = .693147

Using the attached formula:
Ending Amount = Beginning amount * e ^(-lambda * time)
where "e" = 2.718281828
Ending Amount = 180 * e^(-.693147 * time)
Let's test this for 2 hours
Ending Amount = 180 * e^(-.693147 * 2)
Ending Amount = 180 * e^ -1.386294
Ending Amount = 180 * .25
Ending Amount = 45

Answer 2

Answer: $y=180(0.5)^x$

Step-by-step explanation:

The general exponential function is written as :-

$y=Ab^x--------------(i)$

,where y is the amount of the material after t years , A is the initial amount and b is the multiplicative rate of growth or decay.

From the given table , f(0)=180

Thus the  initial value of radioactive material  =180

Also, the decay factor is the ratio of the consecutive terms :

$b=\dfrac{90}{180}=0.5$

Now, put A = 180 and b =0.5 in (i), we get

$y=180(0.5)^x$

Therefore, the exponential function best represents the relationship between f(t) and t is $y=180(0.5)^x$