Download ALGEBRA 2 WKST

AP
Worksheet
Section 6.4
Name ________________________________________hr 4
Date ____/____/____ Score:________/
= ________%
Use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid.
1.
dy y
 and y = 2 when x = 2
dx x
2.
dy
 cos 2 y and y = 0 when x = 0
dx
3.
dy
 e x  y and y = 2 when x = 0
dx
4.
dy
 2xy 2 and y = 0.25 when x = 1
dx
Find the solution of the differential equation dy/dt = ky, k is a constant, that satisfies the given conditions.
5. k = -0.5, y(0) = 200
6. y(0) = 60, y(10) = 30
Complete the table for an investment if interest is compounded continuously.
Initial Deposit ($)
Annual Rate (%)
Doubling Time (yr)
7.
8.
9.
10.
1000
2000
AMOUNT IN 30 YR
($)
8.6
15
5.25
1200
7.
8.
9.
10.
2898.44
10,405.37
Find the amount of time required for a $2000 investment to double if the annual interest rate r is compounded (a)
annually, (b) monthly, (c) quarterly, and (d) continuously.
11. r = 4.75%
12. r = 8.25%
13. The radioactive decay of Sm-151 (an isotope of samarium) can be modeled by the differential equation
dy / dt  0.0077 y , where t is measured in years. Find the half-life of Sm-151.
14. A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially
with time. At the end of 3 h there are 10,000 bacteria. At the end of 5h there are 40,000 bacteria. How many bacteria are
present initially?
15. Find the exponential function y  y 0 e kt whose graph passé through the two points.
16. An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at 65F.
After 10 min, the beam warmed to 35F and after another 10 min its temperature was 50F. Use Newton’s Law of cooling
to estimate the beam’s initial temperature.
17. A temperature probe is removed from a cup of coffee and placed in water whose temperature (Ts ) is 10C. The data
in the table was collected over the next 30 sec with a CBL temperature probe.
Time (sec)
T(C)
(T)- (Ts ) (C)
2
80.47
70.47
5
69.39
59.39
10
49.66
39.66
15
35.26
25.26
20
28.15
18.15
25
23.56
13.56
30
20.62
10.62
(a) Find an exponential regression equation for the ( t , T  Ts ) data.
(b) Use the regression equation in part (a) to find a model for the (t,T) data. Superimpose the graph of the model on a
scatter plot of the (t,T) data.
(c) Estimate when the temperature probe will read 12C.
(d) Estimate the coffee’s temperature when the temperature probe was removed.
18.The charcoal from a tree killed in a volcanic eruption that formed Crater Lake in Oregon contained 44.5% of the
carbon-14 found in living matter. About how old is Crater Lake?
19. A savings account earning compound interest triples in value in 10 years. How long will it take for the original
investment to quadruple?
20. In some chemical reactions the rate at which the amount of a substance changes with time is proportional to the
amount present. For the change of -glucono lactone into gluconic acid, for example,
dy
 0.6 y when y is measured in
dt
hours. If there are 100 grams of a -glucono lactone present when t = 0, how many grams will be left after the first hour?