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Eureka Math, A Story of Functionsยฎ
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Eureka Mathโข Homework Helper
2015โ2016
Algebra IModule 3
Lessons 1โ7
2015-16 M3 A Story of Functions ALGEBRA I
Lesson 1: Integer SequencesโShould You Believe in Patterns?
Lesson 1: Integer SequencesโShould You Believe in Patterns?
Generating Terms of a Sequence
1. Consider a sequence given by the formula ๐๐(๐๐) = 12 โ 7๐๐ starting with ๐๐ = 1. Generate the first 5terms of the sequence.
๐๐(๐๐) = ๐๐๐๐ โ ๐๐(๐๐) = ๐๐
๐๐(๐๐) = ๐๐๐๐ โ ๐๐(๐๐) = โ๐๐
๐๐(๐๐) = ๐๐๐๐ โ ๐๐(๐๐) = โ๐๐
๐๐(๐๐) = ๐๐๐๐ โ ๐๐(๐๐) = โ๐๐๐๐
๐๐(๐๐) = ๐๐๐๐ โ ๐๐(๐๐) = โ๐๐๐๐
The first five terms of the sequence are ๐๐, โ๐๐, โ๐๐, โ๐๐๐๐, โ๐๐๐๐.
Writing a Formula for a Sequence
2. Consider the following sequence that follows a times 12
pattern: 1, 12 , 1
4 , 18 , โฆ.
a. Write a formula for the ๐๐P
th term of the sequence. Be sure to specify what value of ๐๐ your formula starts with.
๐๐(๐๐) = ๏ฟฝ๐๐๐๐๏ฟฝ๐๐โ๐๐
starting with ๐๐ = ๐๐
I can check my formula by using it to generate the first few terms of the sequence.
๐๐(1) = ๏ฟฝ12๏ฟฝ1โ1
= ๏ฟฝ12๏ฟฝ0
= 1
๐๐(2) = ๏ฟฝ12๏ฟฝ2โ1
= ๏ฟฝ12๏ฟฝ1
=12
To find the first five terms of the sequence, I can replace ๐๐ with the numbers 1, 2, 3, 4, and 5.
I see that this sequence has a โminus 7โ pattern. I could use this pattern to continue generating terms in the sequence.
I know that my formula can start with any value of ๐๐ but that the convention is to start with ๐๐ = 1.
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Homework Helper
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
2015-16
2015-16
M3 ALGEBRA I
Lesson 1: Integer SequencesโShould You Believe in Patterns?
b. Using the formula, find the 10P
th term of the sequence.
๐๐(๐๐๐๐) = ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐๐โ๐๐
= ๏ฟฝ๐๐๐๐๏ฟฝ๐๐
= ๐๐๐๐๐๐๐๐
c. Graph the four terms of the sequence as ordered pairs ๏ฟฝ๐๐,๐๐(๐๐)๏ฟฝ on a coordinate plane.
Since my formula starts with ๐๐ = 1, I can find the 10th term by replacing ๐๐ with 10.
The ordered pair for the first term is (1, 1). The ordered pair for the second term is
๏ฟฝ2, 12๏ฟฝ, and so on.
2
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 2: Recursive Formulas for Sequences
Lesson 2: Recursive Formulas for Sequences
Generating Terms of a Sequence When Given a Recursive Formula
List the first five terms of each sequence.
1. ๐๐(๐๐) = โ3๐๐(๐๐ โ 1) and ๐๐(1) = 2 for ๐๐ โฅ 2
๐๐(๐๐) = ๐๐
๐๐(๐๐) = โ๐๐๐๐(๐๐ โ ๐๐) = โ๐๐๐๐(๐๐) = โ๐๐(๐๐) = โ๐๐
๐๐(๐๐) = โ๐๐๐๐(๐๐ โ ๐๐) = โ๐๐๐๐(๐๐) = โ๐๐(โ๐๐) = ๐๐๐๐
๐๐(๐๐) = โ๐๐๐๐(๐๐ โ ๐๐) = โ๐๐๐๐(๐๐) = โ๐๐(๐๐๐๐) = โ๐๐๐๐
๐๐(๐๐) = โ๐๐๐๐(๐๐ โ ๐๐) = โ๐๐๐๐(๐๐) = โ๐๐(โ๐๐๐๐) = ๐๐๐๐๐๐
The first five terms of the sequence are ๐๐,โ๐๐,๐๐๐๐,โ๐๐๐๐,๐๐๐๐๐๐.
2. ๐๐๐๐+1 = ๐๐๐๐ + 2๐๐ + 1 where ๐๐1 = 1 for ๐๐ โฅ 1
๐๐๐๐ = ๐๐
๐๐๐๐ = ๐๐๐๐ + ๐๐(๐๐) + ๐๐ = ๐๐ + ๐๐ + ๐๐ = ๐๐
๐๐๐๐ = ๐๐๐๐ + ๐๐(๐๐) + ๐๐ = ๐๐ + ๐๐ + ๐๐ = ๐๐
๐๐๐๐ = ๐๐๐๐ + ๐๐(๐๐) + ๐๐ = ๐๐ + ๐๐ + ๐๐ = ๐๐๐๐
๐๐๐๐ = ๐๐๐๐ + ๐๐(๐๐) + ๐๐ = ๐๐๐๐ + ๐๐ + ๐๐ = ๐๐๐๐
The first five terms of the sequence are ๐๐,๐๐,๐๐,๐๐๐๐,๐๐๐๐.
I can find the next term in the sequence by starting with ๐๐ = 2.
The subscripts in this notation represent the term number just like the values in the parentheses did in the formula above.
I replaced ๐๐ with 2 in the formula to find the second term.
I replaced ๐๐ with 1 in the formula to find the second term.
I see that this is a recursive formula. To find any term in the sequence, I need to use the previous term.
This means that the first term of the sequence is 2. I can use that to get started.
This sequence is valid for integers greater than or equal to 1.
3
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 2: Recursive Formulas for Sequences
Writing a Recursive Formula for a Sequence
3. Write a recursive formula for the sequence that has an explicit formula ๐๐(๐๐) = 4๐๐ โ 2 for ๐๐ โฅ 1.
๐๐(๐๐) = ๐๐(๐๐) โ ๐๐ = ๐๐
๐๐(๐๐) = ๐๐(๐๐) โ ๐๐ = ๐๐
๐๐(๐๐) = ๐๐(๐๐) โ ๐๐ = ๐๐๐๐
๐๐(๐๐) = ๐๐(๐๐) โ ๐๐ = ๐๐๐๐
๐๐(๐๐ + ๐๐) = ๐๐(๐๐) + ๐๐ where ๐๐(๐๐) = ๐๐ and ๐๐ โฅ ๐๐
4. The bacteria culture has an initial population of 100 and it quadruples in size every hour.
This sequence has a โtimes ๐๐โ pattern: ๐๐๐๐๐๐,๐๐๐๐๐๐,๐๐๐๐๐๐๐๐,๐๐๐๐๐๐๐๐
๐ฉ๐ฉ๐๐+๐๐ = ๐๐๐ฉ๐ฉ๐๐ where ๐ฉ๐ฉ๐๐ = ๐๐๐๐๐๐ and ๐๐ โฅ ๐๐
I can use subscripts or parentheses like ๐ต๐ต(๐๐ + 1) to name the sequence.
It might be helpful to generate the first few terms of the sequence.
I see that this sequence is following a โplus 4โ pattern.
Each term in the sequence is 4 times the previous one.
400 = 4 โ 100 1600 = 4 โ 400
6400 = 4 โ 1600 Noticing this pattern helps me write the recursive formula.
4
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 3: Arithmetic and Geometric Sequences
Lesson 3: Arithmetic and Geometric Sequences
Identifying Sequences as Arithmetic or Geometric
1. List the first five terms of the sequence given below, and identify it as arithmetic or geometric.
๐ด๐ด(๐๐ + 1) = โ3 โ ๐ด๐ด(๐๐) for ๐๐ โฅ 1 and ๐ด๐ด(1) = 2
๐จ๐จ(๐๐) = ๐๐
๐จ๐จ(๐๐) = โ๐๐ โ ๐จ๐จ(๐๐) = โ๐๐ โ ๐๐ = โ๐๐
๐จ๐จ(๐๐) = โ๐๐ โ ๐จ๐จ(๐๐) = โ๐๐ โ โ๐๐ = ๐๐๐๐
๐จ๐จ(๐๐) = โ๐๐ โ ๐จ๐จ(๐๐) = โ๐๐ โ ๐๐๐๐ = โ๐๐๐๐
๐จ๐จ(๐๐) = โ๐๐ โ ๐จ๐จ(๐๐) = โ๐๐ โ โ๐๐๐๐ = ๐๐๐๐๐๐
This sequence is geometric.
2. Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value.
15, 9, 3,โ3,โ9, โฆ
This sequence is arithmetic.
๐๐(๐๐ + ๐๐) = ๐๐(๐๐)โ ๐๐ for ๐๐ โฅ ๐๐ and ๐๐(๐๐) = ๐๐๐๐
Writing the Explicit Form of an Arithmetic or Geometric Sequence
3. Consider the arithmetic sequence 15, 9, 3,โ3,โ9, โฆ. a. Find an explicit form for the sequence in terms of ๐๐.
๐๐(๐๐) = ๐๐๐๐ + (๐๐ โ ๐๐) โ โ๐๐ = โ๐๐๐๐ + ๐๐๐๐ for ๐๐ โฅ ๐๐
I was given a recursive formula and the first term, ๐ด๐ด(1). I can use the first term to find the second term.
I see that each term in the sequence is the product of the previous term and โ3.
I see that each term in the sequence is the sum of the previous term and โ6.
I need to identify the pattern. To find the second term, I need to subtract 6 one time. To find the third term, I need to subtract 6 two times. To find the ๐๐th term, I need to subtract 6 (๐๐ โ 1) times.
5
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 3: Arithmetic and Geometric Sequences
b. Find the 80P
th term.
๐๐(๐๐๐๐) = โ๐๐(๐๐๐๐) + ๐๐๐๐ = โ๐๐๐๐๐๐
4. Find the common ratio and an explicit form for the following geometric sequence.
108, 36, 12, 4, ...
๐๐ = ๐๐๐๐๐๐๐๐๐๐ = ๐๐๐๐
๐๐๐๐ = ๐๐๐๐๐๐ = ๐๐
๐๐
๐๐(๐๐) = ๐๐๐๐๐๐๏ฟฝ๐๐๐๐๏ฟฝ๐๐โ๐๐
5. The first term in an arithmetic sequence is 2, and the 5P
th term is 8. Find an explicit form for the arithmetic sequence.
๐๐ = ๐๐ + ๐๐ โ ๐ ๐ ๐๐๐๐
= ๐ ๐
๐๐(๐๐) = ๐๐ + (๐๐ โ ๐๐) โ ๐๐๐๐ = ๐๐๐๐๐๐ + ๐๐
๐๐
I can find the common ratio, ๐๐, by dividing any two successive terms.
I need to identify the pattern. To find the second term, I
need to multiply by 13
one
time. To find the third term, I
need to multiply by 13
two
times. To find the ๐๐th term, I
need to multiply by 13
(๐๐ โ 1)
I can check my formula by finding a term in the sequence.
๐๐(4) = 108 ๏ฟฝ13๏ฟฝ
4โ1= 108 ๏ฟฝ1
3๏ฟฝ3
= 4
The fourth term of the sequence is 4.
I can check my formula by finding a term in the sequence.
๐๐(5) = 32
(5) + 12
= 8
The fifth term of the sequence is 8.
To find the 80th term, I need to find ๐๐(80) using the explicit form.
2, , , ,8 I need to find some number, ๐๐, that when added to the first term 4 times results in the fifth term.
6
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 4: Why Do Banks Pay YOU to Provide Their Services?
Lesson 4: Why Do Banks Pay YOU to Provide Their Services?
Calculations Involving Simple Interest
1. $800 is invested at a bank that pays 5% simple interest. Calculate the amount of money in the account after 12 years.
๐ฐ๐ฐ(๐๐) = ๐ท๐ท๐ท๐ท๐๐
๐ฐ๐ฐ(๐๐๐๐) = ๐๐๐๐๐๐(๐๐.๐๐๐๐)(๐๐๐๐)
๐ฐ๐ฐ(๐๐๐๐) = ๐๐๐๐๐๐
After ๐๐๐๐ years, the account will have $๐๐,๐๐๐๐๐๐.
Calculations Involving Compound Interest
2. $800 is invested at a bank that pays 5% interest compounded annually. Calculate the amount of money in the account after 12 years.
๐ญ๐ญ๐ญ๐ญ = ๐ท๐ท(๐๐ + ๐ท๐ท)๐๐
๐ญ๐ญ๐ญ๐ญ = ๐๐๐๐๐๐(๐๐ + ๐๐.๐๐๐๐)๐๐๐๐ โ ๐๐,๐๐๐๐๐๐.๐๐๐๐
After ๐๐๐๐ years, the account will have $๐๐,๐๐๐๐๐๐.๐๐๐๐.
I know that simple interest means that interest is earned only on the original investment amount.
I can use this formula to calculate the interest. ๐๐ is the principal amount, and ๐๐ is the interest rate in decimal form.
I know that compound interest means that each time interest is earned, it becomes part of the principal.
I can use this formula to calculate the future value ๐๐ years after investing.
7
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 5: The Power of Exponential Growth
Lesson 5: The Power of Exponential Growth
1. In the year 2000, a total of 768, 586 high school students took an Advanced Placement (AP) exam. Since the year 2000, the number of high school students who take an AP exam has increased at an approximate rate of 9% per year. a. What explicit formula models this situation?
๐๐(๐๐) = ๐๐๐๐๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐)๐๐,
where ๐๐ represents the number of years since ๐๐๐๐๐๐๐๐.
b. If this trend continues, predict the number of high school students who will take an AP exam in the year 2020.
๐๐(๐๐๐๐) = ๐๐๐๐๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐)๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐๐๐๐๐.๐๐๐๐๐๐
If this trend continues, approximately ๐๐,๐๐๐๐๐๐,๐๐๐๐๐๐ students will take an AP exam in the year ๐๐๐๐๐๐๐๐.
2. Jackie decided to start a savings plan where she deposited $0.01 in a jar on day one, $0.03 on day two, $0.09 on day three, and so on, tripling the amount she saved each day. After how many days of following this plan would the amount she deposited in the jar exceed $10,000? Be sure to include an explicit formula with your response.
๐จ๐จ(๐๐) = ๐๐.๐๐๐๐(๐๐)๐๐โ๐๐ for ๐๐ โฅ ๐๐
๐จ๐จ(๐๐๐๐) = ๐๐.๐๐๐๐(๐๐)๐๐๐๐โ๐๐ = ๐๐.๐๐๐๐(๐๐)๐๐๐๐ = ๐๐๐๐๐๐๐๐.๐๐๐๐
๐จ๐จ(๐๐๐๐) = ๐๐.๐๐๐๐(๐๐)๐๐๐๐โ๐๐ = ๐๐.๐๐๐๐(๐๐)๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐.๐๐๐๐
On day ๐๐๐๐ of the savings plan, the amount she deposited would exceed $๐๐๐๐,๐๐๐๐๐๐.
Since ๐ก๐ก represents years since 2000, I need to evaluate ๐๐(20).
In this formula, I am starting with ๐ก๐ก = 0 (the year 2000).
I see that the amount she saves each day forms a geometric sequence with a โtimes 3โ pattern.
I can check my formula. Day 1: ๐ด๐ด(1) = 0.01(3)1โ1 = 0.01 Day 2: ๐ด๐ด(2) = 0.01(3)2โ1 = 0.03 Day 3: ๐ด๐ด(3) = 0.01(3)3โ1 = 0.09
I used trial and error to find the answer.
This is an example of exponential growth, so I need an explicit formula for a geometric sequence to model this situation.
8
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 6: Exponential GrowthโU.S. Population and World Population
Lesson 6: Exponential GrowthโU.S. Population and World
Population
Stuart plans to deposit $1,000 into a savings account. His bank offers two different types of savings accounts.
Option A pays a simple interest rate of 3.2% per year. Option B pays a compound interest rate of 2.8% per year, compounded monthly.
a. Write an explicit formula for the sequence that models the balance in Stuartโs account ๐ก๐ก years after he deposits the money if he chooses option A.
๐จ๐จ(๐๐) = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐
b. Write an explicit formula for the sequence that models the balance in Stuartโs account ๐๐ months after he deposits the money if he chooses option B.
๐ฉ๐ฉ(๐๐) = ๐๐๐๐๐๐๐๐๏ฟฝ๐๐ + ๐๐.๐๐๐๐๐๐๐๐๐๐ ๏ฟฝ
๐๐
c. Which option is represented with a linear model? Why?
Option A is represented with a linear model because there is a constant rate of change each year.
d. Which option is represented with an exponential model? Why?
Option B is represented with an exponential model because there is a constant ratio of change each month.
I know that simple interest means that the same amount of interest will be added each year. I can use the formula ๐ผ๐ผ = ๐๐๐๐๐ก๐ก to write an expression for the total interest.
Since the interest is compounded monthly, I need to divide the annual interest rate by 12 to find the interest rate per month.
9
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 6: Exponential GrowthโU.S. Population and World Population
e. Approximately how long will it take Stuart to double his money if he chooses option A? Option B?
๐จ๐จ(๐๐) = ๐๐๐๐๐๐๐๐+ ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐ ๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐+ ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐
๐๐ = ๐๐๐๐.๐๐๐๐
๐ฉ๐ฉ(๐๐) = ๐๐๐๐๐๐๐๐๏ฟฝ๐๐ +๐๐.๐๐๐๐๐๐๐๐๐๐
๏ฟฝ๐๐
๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐๏ฟฝ๐๐ +๐๐.๐๐๐๐๐๐๐๐๐๐
๏ฟฝ๐๐
๐๐ โ ๐๐๐๐๐๐
If he chooses option A, it will take him ๐๐๐๐ years to double his money. If he chooses option B, it will take him ๐๐๐๐ years and ๐๐ months to double his money.
f. How should Stuart decide between the two options?
If he is going to invest for a short amount of time (fewer than ๐๐๐๐ years), he should choose option A. If he is going to invest for a long amount of time (๐๐๐๐ years or longer), he should choose option B.
๐จ๐จ(๐๐๐๐) = ๐๐๐๐๐๐๐๐+ ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐๐๐ = ๐๐๐๐๐๐๐๐
๐ฉ๐ฉ(๐๐๐๐๐๐) = ๐๐๐๐๐๐๐๐๏ฟฝ๐๐ + ๐๐.๐๐๐๐๐๐๐๐๐๐ ๏ฟฝ
๐๐๐๐๐๐โ ๐๐๐๐๐๐๐๐.๐๐๐๐
I can solve this equation for ๐ก๐ก.
At 10 years (120 months), the balance in the account for option B is larger than the balance in the account for option A.
I need to use trial and error to find ๐๐.
I need to determine when ๐ด๐ด(๐ก๐ก) and ๐ต๐ต(๐ก๐ก) will equal $2000.
10
ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 7: Exponential Decay
Lesson 7: Exponential Decay
1. Since 1950, the population of Detroit has been decreasing. The population of Detroit (in millions) can be modeled by the following formula:
๐๐(๐ก๐ก) = 1.85(0.985)๐ก๐ก, where ๐ก๐ก is the number of years since 1950.
a. According to the model, what was the population of Detroit in 1950?
๐ท๐ท(๐๐) = ๐๐.๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐ = ๐๐.๐๐๐๐
In 1950, the population of Detroit was approximately ๐๐.๐๐๐๐ million.
b. Complete the following table, and then graph the points ๏ฟฝ๐ก๐ก,๐๐(๐ก๐ก)๏ฟฝ.
๐๐ ๐ท๐ท(๐๐)
0 ๐๐.๐๐๐๐
10 ๐๐.๐๐๐๐
20 ๐๐.๐๐๐๐
30 ๐๐.๐๐๐๐
40 ๐๐.๐๐๐๐
50 ๐๐.๐๐๐๐
60 ๐๐.๐๐๐๐
I need to find ๐๐(0) since 1950 corresponds to ๐ก๐ก = 0.
I see that the points form an exponential decay curve.
I can see that this is an exponential decay model because ๐๐ < 1 in the formula ๐๐(๐ก๐ก) = ๐๐(๐๐)๐ก๐ก.
Number of years since 1950, ๐๐
Popu
latio
n (in
mill
ions
), ๐ท๐ท
(๐๐)
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ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions
2015-16
2015-16
M3 ALGEBRA I
Lesson 7: Exponential Decay
c. If this trend continues, estimate the year in which the population of Detroit will be less than 500,000.
๐ท๐ท(๐๐๐๐) = ๐๐.๐๐๐๐(๐๐.๐๐๐๐๐๐)๐๐๐๐ โ ๐๐.๐๐๐๐๐๐
If this trend continues, the population of Detroit will be less than ๐๐๐๐๐๐,๐๐๐๐๐๐ by the year 2037.
2. A Christmas tree farmer has 6,000 firs on his farm. Each Christmas, he plans to cut down 12% of his trees. a. Write a formula to model the number of trees on his farm each year.
๐ต๐ต(๐๐) = ๐๐๐๐๐๐๐๐(๐๐ โ ๐๐.๐๐๐๐)๐๐ = ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐)๐๐, where ๐๐ represents the number of years.
b. If he does not plant any new trees, how many trees will he have on his farm in 15 years?
๐ต๐ต(๐๐๐๐) = ๐๐๐๐๐๐๐๐(๐๐.๐๐๐๐)๐๐๐๐ โ ๐๐๐๐๐๐.๐๐๐๐๐๐
After ๐๐๐๐ years, the farmer will have approximately ๐๐๐๐๐๐ trees on his farm.
I can use trial and error or the table feature on a graphing calculator to find the answer.
If the farmer cuts down 12% of the trees each year, then 88% of the trees are remaining.
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ยฉ 2015 Great Minds eureka-math.org ALG I-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Functions