Exponential Application

Question

A scientist is measuring a material that she thinks may be radioactive. She begins with 300 grams, and measures that after 4 hours, she has 240 grams remaining.

a) Find an exponential model of the form A(t)= A0e^kt that will model this material.

b)Find the half-life of the material.

d) Determine how much will be left after 1 day.

Answer 1

(part A)

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f(t) = K * e^(kt)

f(0) = 300 ---> k = 300

so f(t) = 300 * exp(kt)

(4, 240) ---> 240 = 300*(e^4k)

240/300 = (e^4k)

4/5 = e^(4k)

ln 4/5 = 4k

(ln 4/5)/4 = k

(1/4)(ln 4/5) = k

f(t) = 300 * exp( (ln 4/5) (t/4))

= 300 * (exp (ln 4/5) )^(t/4)

= 300 * (4/5)^(t/4)

part (B)

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Half life : 1/2 = (4/5)^(t/4)

ln(1/2) = (t/4) ln (4/5)

-ln2 = (t/4) (ln 4 - ln 5)

-4 ln 2 / (ln 4 - ln 4) = t

part (C)

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f(24) = 300 * (4/5)^6 = 78.6432