The quantity, in grams, of a radioactive substance present after t months is given by the
function Q(t) = 1000(5)^(−0.3t)
(a) How much of the substance will be present after 6 months?
(b) How long until there is only 8 grams of the substance remaining?
To answer these questions, we need to substitute the given values into the function Q(t) and solve for the unknown quantity. Let's calculate each question step by step:
(a) How much of the substance will be present after 6 months?
To find the quantity of the substance after 6 months, we plug t = 6 into the Q(t) function:
Q(6) = 1000(5)^(-0.3*6)
First, calculate -0.3*6:
-0.3*6 = -1.8
Now, substitute this value into the equation:
Q(6) = 1000(5)^(-1.8)
To further simplify, calculate 5^(-1.8):
5^(-1.8) ≈ 0.079432823
Finally, multiply this value by 1000:
Q(6) = 1000 * 0.079432823 ≈ 79.43 grams
Therefore, after 6 months, approximately 79.43 grams of the substance will be present.
(b) How long until there is only 8 grams of the substance remaining?
To find the time it takes for there to be 8 grams remaining, we need to set Q(t) equal to 8 and solve for t:
8 = 1000(5)^(-0.3t)
Divide both sides of the equation by 1000:
0.008 = (5)^(-0.3t)
Now, take the logarithm of both sides (base 5):
log5(0.008) = -0.3t
Simplify the logarithm using a calculator:
t ≈ log5(0.008) / -0.3
Evaluate the expression using a calculator to find t:
t ≈ -8.289
Therefore, it will take approximately -8.289 months until there is only 8 grams remaining. Since time cannot be negative, we can conclude that there is no time at which there will be 8 grams of the substance remaining.
In summary, after 6 months, approximately 79.43 grams of the substance will be present, and there is no time at which there will be exactly 8 grams remaining.