Help with exponential decay problem please?

Question

A student was asked to fill in a table like the one below, where the quantity of A(t) decays exponentially, and t is measured in days

T 0 50 100 150 200 250 300
A(t) 800 400 200 100

The student completes the table so:

T 0 50 100 150 200 250 300
A(t) 800 600 400 300 200 150 100

Using this reasoning, 600 is halfway between 800 & 400 . By looking a the given values of A(t), the half-life is 100 days so 100 days later the value should be half of 600 which is 300 and so on

In what ways did the student reason correctly? In what ways did the student reason incorrectly. What is the function described in the student’s table?

Fill in the table with the correct values. Did the student overestimate or underestimate. Sketch a graph of the function

Answer 1

Your spacing is out of whack so it's a bit hard to see what you mean. I'm going to assume the initial value at t=0 was 800, and the half life where A=400 was at t=100 days.

A straight simply decay formula is of the form A(t) = A_o*e^(kt)

from the given data, A(0) = 800 = A_o*e^(k*0) = A_o*e^0 = A_o*1

Therefore, A_o = 800

Since you say the given half life is 100 days, you now have a second equation for your second unknown.

A(100) = 400 = 800*e^(100k)
0.5 = e^100k
ln(0.5) = 100k

k = ln(0.5)/100

So your equation is A(t) = 800*e^(ln(0.5)/100*t)

Now you can fill in the missing values. Make sure in these types of problems your units match. You can also explain why the "student" should pack his/her bags and leave school for being a dummy.

Answer 2

At t = 0, A(t) = 800, right?
At t = 100, A(t) = 400, yes?
At t = 200, A(t) = 200, am I right so far?
At t = 300, A(t) = 100

If you look at t = 0 and t = 200, you can see that "halfway" reasoning is half-assed, because 400 is not halfway between 800 and 200. What is 400? Not the arithmetic mean of 800 and 200, but the GEOMETRIC mean, the square root of 800 * 200.

So the other entries must be geometric means.

At t = 50, A(t) = sqrt(800 * 400)
At t = 150, A(t) = sqrt(400 * 200)
At t = 250, A(t) = sqrt(200 * 100)

The function is

A(t) = 800e^(-kt)

Now we have to find k.

At t = 100, A(t) = 400 = 800e^(-100k)

1/2 = e^(-100k)
ln(1/2) = -100k
-0.693 = -100k
k = .00693

Answer 3

It is the same difference as between simple and compound interest. The student used linear approximation to calculate the half life. (I assume the measured values are meant to be at 0, 100, 200, and 300). Graphically this would give a piecewise linear curve approximating the actual exponential curve. The student wants to use the geometric mean (geometric mean of a and b is sqrt(ab), arithmatic mean is (a + b)/2) to get the actual values.