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)