Engineering Calculus
Homework # 3
Homework # 3
7.4 # 8: (3 pts, p. 528) Strontium-90 has a half life of 28 days.
(a) A sample has a mass of 50 mg initially. Find a formula for the the mass remaining after
t days.
(b) Find the mass remaining after 40 days.
(c) How long does it take the sample to decay to a mass of 2 mg.
(d) Sketch a graph of the mass function.
(a): The formula will be similar to m(t) = m0 ekt . To find the value of k (using days for the
unit of time), assume that a starting mass of m0 will be reduced to a mass of 21 m0 in 28 days.
= m0 e28k
ln 2
k=−
28
m(t) = 50e−t ln 2/28 ≈ 50e−0.024755256t
1
2 m0
With t in days
m(t) = 50 · 2−t/28
(b): For the mass after 40 days:
m(40) = 50 · 2−40/28 = 50 · 2−10/7 ≈ 50 · 0.3714985723 = 18.5749286142 mg
(c): The amount of time to decay to 2 mg:
2 = 50e−t ln 2/28
28 ln 25
t=
≈ 130.0279733137 days
ln 2
(d): A graph of the mass function:
50
Mass in mg
40
30
20
10
0
0
20
40
60
Time in Days
80
100
8.2 # 12: (3 pts, p. 572) Determine whether the geometric series is convergent or divergent.
If it is convergent, find its sum.
1
Engineering Calculus
Homework # 3
4+3+
The ratio between consecutive terms is
3
4
S=
9
4
27
16
+
+ ···
so this geometric series converges. Its sum is:
4
1−
3
4
= 16
8.3 # 2: (4 pts, p. 583) Suppose f is a continuous positive decreasing function for x ≥ 1 and
an = f (n). By drawing a picture, rank the following three quantities in increasing order:
∫
5
∑
6
f (x) dx
1
6
∑
ai
i=1
Here are three pictures. The values in greatest to least are (b)
then (c)
6
∑
5
∑
i=1
ai .
i=2
Section 8.3 # 2 (a)
y
4
3
2
1
x
1
2
3
4
5
6
7
6
7
Section 8.3 # 2 (b)
y
4
3
2
1
x
1
2
3
4
2
ai
i=2
5
∫
6
ai , (a)
f (x) dx, and
1
Engineering Calculus
Homework # 3
Section 8.3 # 2 (c)
y
4
3
2
1
x
1
2
3
4
3
5
6
7