Question

The mass of a sample of the chemical element einsteinium-253253253 after it is initially measured is represented by the following table:

Time (weeks) Mass (grams)

000 400400400

333 201201201

666 989898

999 505050

121212 242424

151515 121212

Which model for M(t)M(t)M, left parenthesis, t, right parenthesis, the mass of the sample ttt weeks after it's initially measured, best fits the data?

Choose 1 answer:

Choose 1 answer:


(Choice A)

A

M(t)=400-12\cdot tM(t)=400−12⋅tM, left parenthesis, t, right parenthesis, equals, 400, minus, 12, dot, t


(Choice B)

B

M(t)=400-200\cdot tM(t)=400−200⋅tM, left parenthesis, t, right parenthesis, equals, 400, minus, 200, dot, t


(Choice C)

C

M(t)=400\cdot(0.5)^tM(t)=400⋅(0.5)

t

M, left parenthesis, t, right parenthesis, equals, 400, dot, left parenthesis, 0, point, 5, right parenthesis, start superscript, t, end superscript


(Choice D)

D

M(t)=400\cdot(0.79)^tM(t)=400⋅(0.79)

t

M, left parenthesis, t, right parenthesis, equals, 400, dot, left parenthesis, 0, point, 79, right parenthesis, start superscript, t, end superscript

Answer 1

M(t) = 400•(0.79)^t

Explanation:

As we can see, the change in mass is not uniform linearly

So it will be better if we used an exponential representation

The general form for an exponential representation is:

M(t) = I(1-r)^t

where r is the rate of decay, I is the initial value and t is the time in weeks with M(t) being the mass

Let us use any two points on the table

Thus, we have it that;

50 = 400(1-r)^9 •••••(i)

201 = 400(1-r)^3 •••••(ii)

divide i by ii

50/201 = (1-r)^6

0.249 = (1-r)^6

1-r = 6√0.249

1-r = 0.79

r = 1-0.79

r = 0.21

So the exponential equation is;

M(t) = 400•0.79^t