Answer to Question #300376 in Statistics and Probability for dylan

Question

4. [6] The table below gives ²¹⁰Pb activities in a marine sediment core.

Depth, L(cm)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

²¹⁰Pb activity, A_L(Bq kg^-1)

247.0

199.0

160.0

123.0

101.0

78.0

62.0

46.0

38.0

Assume the following equation applies

Where A_L is the activity at depth L (Bq kg^-1), A₀ is the activity at zero depth (L = 0 cm), l is the decay constant (0.031 yr^-1 for ²¹⁰Pb), L is the sediment depth (cm) and S is the sediment accumulation rate (cm yr^-1).

(a) Transform the above equation into straight line form.

(b) By plotting the data estimate the value of A₀ and S.

Answer 1

Let t be the age of a layer. Therefore, $L = tS.$

The activity is $A_L = A_0e^{-tl}.$

So $A_L = A_0e^{-\frac{L}{S}l}.$ It is an exponential curve. If we take logarithm, we'll get

$\ln A_L = \ln A_0 - \frac{L}{S}l = \ln A_0 - L \cdot\dfrac{l}{S}.$ It is a linear function of L.

From a linear plot we get $\ln A_0 = 5.9 \; \Rightarrow A_0 = e^{5.9} = 365$ Bq kg^-1 ,$\frac{l}{S} = 0.475 \Rightarrow S = 0.065$ cm/y.