there's a table which looks like this..

(time)0- (amount)500

0.5- 420

1- 360

1.5- 290

2- 245

2.5- 216 and so on...

I need to find the half life equation that models this
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t Actual  Predicted
0 500 500
0.5 420 420
1 360 353
1.5 290 296
2 245 249
2.5 216 209

We can apply y=ae^(kt).

When t=0, y=a, so a=500. y=500e^(kt).

Now put some values for y and t:

y=420, t=0.5: 420=500e^(0.5k), (420/500)=e^(0.5k)

ln(0.84)=0.5k, k=2ln(0.84)=-0.3487 approx.

This would imply y=500e^(-0.3487t).

Now put t=2, when we expect y=245. But let’s see what the model gives: y=500e^(-0.3487×2)=249 approx. This is about 2% in error.

In the table above the third column shows the prediction according to the model. The predicted values are in error by around 2%.

The predicted half life is when y=250: half-life=-ln(2)/(2ln(0.84))=2 approximately. From the given table, 245 is the actual value when t=2, while the predicted value is 249—close to 250.

If we had used t=2 instead of t=0.50, k=-0.3567. So the predicted half-life would be 1.94, that is, approximately 2, as before. The average of the two k values is -0.3523. This value would give a half-life of 1.96, approximately 2.

 

 

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