power point exponential function applications
TRANSCRIPT
Applications of Exponential Functions
COMPOUND INTEREST
1nt
rA P
n
1. Cyril is being offered with an investment that promises a 10% annual interest compounded once a year. If he plans to invest Php 10, 000, how much is his money at the end of three years?
n
33
33
33
3
3
A (1 )
A (1 )
A 10,000(1 0.10)
A 10,000(1.10)
A 10,000(1.331)
A 13,310
tP r
P r
Given:P = 10,000r = 0.10n= 3
2. Find the amount if Php 3, 000 is invested at 5% annual interest rate for 8 years
(a) Compounded semi-annually
:
3,000
2
0.05
Given
P
n
r
8(2)
8
16
8
16
8
8
8
0.053,000 1
2
3,000 1 0.025
3,000 1.025
3,000 1.4845
4,454
A
A
A
A
A
b. Compounded quarterly
:
3,000
4
0.05
Given
P
n
r
8(4)
8
32
8
32
8
8
8
0.053,000 1
4
3,000 1 0.0125
3,000 1.0125
3,000 1.4881
4,464
A
A
A
A
A
3. A savings account of 5, 000 is placed at 5% per annum. How much is in the account at the end of one year if the interest is:
a. Compounded once a year (n =1)?
1nt
rA P
n
(1)(1)0.05
5000 11
5000(1.05)
5,250
A
The amount if
compounded once a year.
(4)(1)
4
For n = 4
0.05A = 5000 1+
4
= 5000(1.0125)
= 5000(1.050945)
= 5, 254.73
The amount if compounded quarterly.
Compounded quarterly (n = 4)?
Exponential growth
( )toy A r
(1 )ty P r
1. The number of bacteria of a certain type in a certain person’s body doubles every 40 minutes. If 6 are present at 2 A.M., how many will there be at 12 noon?
0
15
( )
6(2)
6(32,768)
196,608
ty A r
y
y
y
2. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of cell subscribers increased by 75 % per year after 1985. How many cell phone subscribers in Centerville last year 2000?
Answer: 1, 260, 131 (rounded value)
EXPONENTIAL DECAY
• When an antibiotic is added to the culture of bacteria, the number of bacteria is reduced by half every 3 hours. If there are 80, 000 bacteria, how many bacteria are left after a day?
0
81
80,0002
312.5
313
xy A r
y
y
y
• A radioactive substance decays so that every year 75% of the previous year’s substance remains. If there are 4 kg of substance at the start of the experiment, how much remains after 3 years?
t
3
3
y = P(1 - r)
= 4 (1 - 0.75)
= 4 ( 0.25)
= 4(0.0156)
= 0.0625 kg
• The number of wolves in the wild in the northern section of a certain country is decreasing at the rate of 3.5 % per year. Your environmental studies class as counted 80 wolves in the area. How many wolves are left after 10 years?
10
1
80(1 0.035)
56.02
56
ty P r
y
y
y
EXERCISES
1. A Php 1,000-deposit is made at a bank that pays 12% interest compounded annually. How much will you have in your account at the end of 10 years?
Given:P = Php 1, 000r = 12 %t = 10 years n = 1Find: Deposited amount after 10 years
Answer: Php 3, 105.85
2. A Php 1,000 deposit is made at a bank that pays 12% interest compounded quarterly. How much will you have in your account at the end of 10 years?
Given:P = Php 1, 000r = 12 % compounded quarterlyt = 10 years Find: Deposited amount after 10 years
Answer: Php 3,262.04
3. Suppose Mitch has Php 1, 000 that she invests in an account that pays 3.5% interest compounded quarterly. How much money does she have at the end of 5 years?
Given:P = Php 1, 000r = 3.5 % compounded quarterlyt = 5 years Find: Deposited amount after 5 years
Answer: Php 1,190.34
4. How much will be on your account after 5 years if you deposited Php 10,000 at 20 % per annum compounded semi annually?
Answer: Php 25, 937. 42
• A population of 200 frogs increases at an annual rate of 22%. How many frogs will there be in 10 years?
Answer: Php 1, 461
• Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players will remain after 5 rounds?
Answer: 4 players
• Suppose the amount in grams of plutonium 241 present in a given sample is determined by the function defined by .
• where t is measured in years. Find the amount present in the sample after the 4 years.
0.053( ) 2.00 tA t e
0.053
0.053(4)
0.212
( ) 2.00
2
2
1.6
tA t e
e
e
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