Page 180 - How To Solve Word Problems In Calculus

3. The function representing the amount of the substance remaining
after t years is Q(t) = Q 0 e −kt . The initial amount, Q 0 = Q(0) =
200.
To determine the value of k, we substitute Q(12) = 180 into
Q(t) = 200e −kt .

Q(12) = 200e −12k

180 = 200e −12k
9
= e −12k
10

9
−12k = ln
10

1 9 1 10
k =− ln = ln
12 10 12 9
10
1 ln )t
The function becomes Q(t) = 200e −( 12 9 . To determine the
half-life, we find the value of t for which Q(t) = 100, half the
original mass.
10
1 ln )t
Q(t) = 200e −( 12 9
1 ln )t
10
100 = 200e −( 12 9
1 1 10
= e −( 12 ln )t
9
2

1 10 1
− ln t = ln
12 9 2
1
−12 ln
2
t = ≈ 78.95 years
10
ln
9
4. The function that governs the decay of the radioactive substance is
ln 2 ln 2
Q(t) = Q 0 e −kt where Q 0 = 10 and k = = .
T 20
ln 2
Q(t) = 10e −( 20 )t
167