Precalculus question

Question

At the beginning of an experiment, a scientist has 272 grams of radioactive goo. After 225 minutes, her sample has decayed to 17 grams. 

What is the half-life of the goo in minutes?     

Find a formula for G(t), the amount of goo remaining at time t. G(t)=    

How many grams of goo will remain after 23 minutes?

Answer 1

To find the half-life of the radioactive goo, we can use the formula:

A = A0 * (1/2)^(t/T)

where A is the amount of radioactive substance remaining after time t, A0 is the initial amount of radioactive substance, T is the half-life, and t is the time elapsed.

Substituting the given values, we get:

17 = 272 * (1/2)^(225/T)

Dividing both sides by 272, we get:

(1/2)^(225/T) = 17/272

Taking the logarithm of both sides with base 1/2, we get:

225/T = log_1/2(17/272)

Simplifying using the change of base formula for logarithms, we get:

T = -225 / log_2(17/272)

Using a calculator, we can approximate the value of T to be approximately 51.55 minutes.

Therefore, the half-life of the radioactive goo is approximately 51.55 minutes.

To find a formula for G(t), the amount of goo remaining at time t, we can use the formula:

G(t) = A0 * (1/2)^(t/T)

Substituting the given values, we get:

G(t) = 272 * (1/2)^(t/51.55)

To find the amount of goo remaining after 23 minutes, we can substitute t = 23 into the formula for G(t):

G(23) = 272 * (1/2)^(23/51.55)

Using a calculator, we can approximate the value of G(23) to be approximately 108.49 grams.

Therefore, after 23 minutes, approximately 108.49 grams of goo will remain.

Answer 2

1/2 = e^rt

.5= e^-.0125t

-.0125t=ln.5

t= -ln.5/.0125= 55.45 minutes = half life time

17 =282e^r225

17/282 = e^225r

ln(17/282) = 225r

r = ln(17/282)/225= about -.0125

G(t) =Pe^-.0125t=272e^-.0125t

G(23)= 272e^-.0125(23)

= about 204 grams after 23 minutes