Answer:
Explanation:
To find the half-life of the radioactive goo in minutes, we can use the formula:
t(1/2) = (ln(2))/k
Where t(1/2) is the half-life, ln represents the natural logarithm, and k is the decay constant.
In this case, we have the initial amount of goo (140 grams) and the amount remaining after a certain time (17.5 grams). We can use this information to find the value of k.
Using the equation G(t) = G(0) * e^(-kt), where G(t) represents the amount of goo remaining at time t, G(0) is the initial amount of goo, e is the base of the natural logarithm, and k is the decay constant, we can substitute the values into the equation:
17.5 = 140 * e^(-k * 240)
To find the value of k, we can divide both sides of the equation by 140:
0.125 = e^(-k * 240)
Now, taking the natural logarithm of both sides:
ln(0.125) = -k * 240
Solving for k:
k = -(ln(0.125))/240
Now that we have the value of k, we can use the half-life formula to find the half-life in minutes:
t(1/2) = (ln(2))/k
Substituting the value of k:
t(1/2) = (ln(2))/(-(ln(0.125))/240)
Simplifying:
t(1/2) = (ln(2)) * (-240/ln(0.125))
t(1/2) ≈ 554.7 minutes
The formula for G(t), the amount of goo remaining at time t, is:
G(t) = G(0) * e^(-kt)
To find the amount of goo remaining after 5 minutes, we can substitute the values into the formula:
G(t) = 140 * e^(-k * 5)
Using the value of k we calculated earlier:
G(t) ≈ 140 * e^(-(-(ln(0.125))/240) * 5)
Simplifying:
G(t) ≈ 140 * e^((ln(0.125))/48)
G(t) ≈ 140 * 0.625
G(t) ≈ 87.5 grams
Therefore, after 5 minutes, approximately 87.5 grams of goo will remain.