154k views
5 votes
At the beginning of an experiment, a scientist has 208 grams of radioactive goo. After 240 minutes, her sample has decayed to 13 grams.

What is the half-tife 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 90 minutes?
You may enter the exact value or round to 2 decimal places.

User Ravi R
by
8.4k points

1 Answer

5 votes

Hello! To calculate the half-life of the radioactive goo, we can use the formula:

⇒ t1/2 = (ln 2)/k

where t1/2 is the half-life, ln represents the natural logarithm, and k is the decay constant.

To solve for k, we can use the formula:

k = (ln (A0/A1))/t

where A0 is the initial amount of radioactive material, A1 is the amount remaining after a given time t, and t is the time elapsed.

Plugging in the values, we get:

k = (ln (208/13))/240

k = 0.0098

Then, plugging in k to the first formula, we get:

t1/2 = (ln 2)/0.0098

t1/2 = 70.8 minutes

Therefore, the half-life of the radioactive goo is approximately 70.8 minutes.

To find the formula for G(t), we can use the following formula:

G(t) = A0 * e^(-kt)

where A0 is the initial amount of radioactive material, k is the decay constant, and t is the time elapsed.

Plugging in the values, we get:

G(t) = 208 * e^(-0.0098 * 240)

G(t) = 13 grams

Therefore, the amount of goo remaining after 90 minutes would be:

G(90) = 208 * e^(-0.0098 * 90)

G(90) = 78.8 grams (rounded to two decimal places)

User Thalsan
by
7.6k points