Math / Algebra

QuestionConsider this scenario: For each year $t$ , the population of a forest of trees is represented by the function $A(t)=97(1.028)^{t}$ . In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=117(1.021)^{t}$ . (Round answers to the nearest whole number.)
Which forest's population is growing at a faster rate? Select an answer population is growing at a faster rate.
Which forest had a greater number of trees initially? By how many? Sellact an answer $\checkmark$ had $\square$ more trees initially.
Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?
In 20 years, Select an answer $\checkmark$ will have $\square$ more trees.
Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 100 years? By how many?
In 100 years, Select an answer $v$ will have $\square$ more trees.

Studdy Solution

STEP 1

What is this asking? We're comparing two forests' tree populations, which are growing exponentially, to see which one grows faster, which one started with more trees, and which one will have more trees after 20 and 100 years. Watch out! Don't mix up the initial population with the growth rate!
A bigger starting number doesn't automatically mean a bigger number later on.

STEP 2

1. Identify Growth Rates
2. Find Initial Populations
3. Calculate Population after 20 Years
4. Calculate Population after 100 Years

STEP 3

Forest A's population function is $A(t) = 97(1.028)^t$ .
The growth factor is $1.028$ .
This means the population increases by $1.028 - 1 = 0.028$ each year.
Converting this to a percentage, we get 0.028 \cdot 100\% = \textbf{2.8%} growth per year.

STEP 4

Forest B's population function is $B(t) = 117(1.021)^t$ .
The growth factor is $1.021$ .
This means the population increases by $1.021 - 1 = 0.021$ each year.
As a percentage, this is 0.021 \cdot 100\% = \textbf{2.1%} growth per year.

STEP 5

Since $2.8\% > 2.1\%$ , Forest A's population is growing at a faster rate.

STEP 6

The initial population is when $t = 0$ .
So, for Forest A, $A(0) = 97(1.028)^0 = 97 \cdot 1 = \textbf{97}$ trees.

STEP 7

For Forest B, $B(0) = 117(1.021)^0 = 117 \cdot 1 = \textbf{117}$ trees.

STEP 8

Forest B initially had more trees.
The difference is $117 - 97 = \textbf{20}$ more trees.

STEP 9

$A(20) = 97(1.028)^{20} \approx 97 \cdot 1.7535 \approx \textbf{170}$ trees.

STEP 10

$B(20) = 117(1.021)^{20} \approx 117 \cdot 1.5223 \approx \textbf{178}$ trees.

STEP 11

Forest B will have more trees after 20 years.
The difference is approximately $178 - 170 = \textbf{8}$ more trees.

STEP 12

$A(100) = 97(1.028)^{100} \approx 97 \cdot 15.6757 \approx \textbf{1521}$ trees.

STEP 13

$B(100) = 117(1.021)^{100} \approx 117 \cdot 8.5772 \approx \textbf{1003}$ trees.

STEP 14

Forest A will have more trees after 100 years.
The difference is approximately $1521 - 1003 = \textbf{518}$ more trees.

STEP 15

Forest A is growing faster at a rate of 2.8% compared to Forest B's 2.1%.
Forest B initially had 20 more trees.
After 20 years, Forest B will have approximately 8 more trees.
After 100 years, Forest A will have approximately 518 more trees.

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Studdy Solution

STEP 1

STEP 2

1. Identify Growth Rates
2. Find Initial Populations
3. Calculate Population after 20 Years
4. Calculate Population after 100 Years

STEP 3

Forest A's population function is $A(t) = 97(1.028)^t$ .
The growth factor is $1.028$ .
This means the population increases by $1.028 - 1 = 0.028$ each year.
Converting this to a percentage, we get 0.028 \cdot 100\% = \textbf{2.8%} growth per year.

STEP 4

Forest B's population function is $B(t) = 117(1.021)^t$ .
The growth factor is $1.021$ .
This means the population increases by $1.021 - 1 = 0.021$ each year.
As a percentage, this is 0.021 \cdot 100\% = \textbf{2.1%} growth per year.

STEP 5

Since $2.8\% > 2.1\%$ , Forest A's population is growing at a faster rate.

STEP 6

The initial population is when $t = 0$ .
So, for Forest A, $A(0) = 97(1.028)^0 = 97 \cdot 1 = \textbf{97}$ trees.

STEP 7

For Forest B, $B(0) = 117(1.021)^0 = 117 \cdot 1 = \textbf{117}$ trees.

STEP 8

Forest B initially had more trees.
The difference is $117 - 97 = \textbf{20}$ more trees.

STEP 9

$A(20) = 97(1.028)^{20} \approx 97 \cdot 1.7535 \approx \textbf{170}$ trees.

STEP 10

$B(20) = 117(1.021)^{20} \approx 117 \cdot 1.5223 \approx \textbf{178}$ trees.

STEP 11

Forest B will have more trees after 20 years.
The difference is approximately $178 - 170 = \textbf{8}$ more trees.

STEP 12

$A(100) = 97(1.028)^{100} \approx 97 \cdot 15.6757 \approx \textbf{1521}$ trees.

STEP 13

$B(100) = 117(1.021)^{100} \approx 117 \cdot 8.5772 \approx \textbf{1003}$ trees.

STEP 14

Forest A will have more trees after 100 years.
The difference is approximately $1521 - 1003 = \textbf{518}$ more trees.

STEP 15

Forest A is growing faster at a rate of 2.8% compared to Forest B's 2.1%.
Forest B initially had 20 more trees.
After 20 years, Forest B will have approximately 8 more trees.
After 100 years, Forest A will have approximately 518 more trees.

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Studdy Solution

STEP 1

STEP 2

1. Identify Growth Rates2. Find Initial Populations3. Calculate Population after 20 Years4. Calculate Population after 100 Years

STEP 3

STEP 4

Forest B's population function is B(t)=117(1.021)tB(t) = 117(1.021)^tB(t)=117(1.021)t.The **growth factor** is 1.0211.0211.021.This means the population increases by 1.021−1=0.0211.021 - 1 = 0.0211.021−1=0.021 each year.As a percentage, this is 0.021 \cdot 100\% = \textbf{2.1%} growth per year.

STEP 5

Since 2.8%>2.1%2.8\% > 2.1\%2.8%>2.1%, Forest A's population is growing at a faster rate.

STEP 6

The initial population is when t=0t = 0t=0.So, for Forest A, A(0)=97(1.028)0=97⋅1=97A(0) = 97(1.028)^0 = 97 \cdot 1 = \textbf{97}A(0)=97(1.028)0=97⋅1=97 trees.

STEP 7

For Forest B, B(0)=117(1.021)0=117⋅1=117B(0) = 117(1.021)^0 = 117 \cdot 1 = \textbf{117}B(0)=117(1.021)0=117⋅1=117 trees.

STEP 8

Forest B initially had more trees.The difference is 117−97=20117 - 97 = \textbf{20}117−97=20 more trees.

STEP 9

A(20)=97(1.028)20≈97⋅1.7535≈170A(20) = 97(1.028)^{20} \approx 97 \cdot 1.7535 \approx \textbf{170}A(20)=97(1.028)20≈97⋅1.7535≈170 trees.

STEP 10

B(20)=117(1.021)20≈117⋅1.5223≈178B(20) = 117(1.021)^{20} \approx 117 \cdot 1.5223 \approx \textbf{178}B(20)=117(1.021)20≈117⋅1.5223≈178 trees.

STEP 11

Forest B will have more trees after 20 years.The difference is approximately 178−170=8178 - 170 = \textbf{8}178−170=8 more trees.

STEP 12

A(100)=97(1.028)100≈97⋅15.6757≈1521A(100) = 97(1.028)^{100} \approx 97 \cdot 15.6757 \approx \textbf{1521}A(100)=97(1.028)100≈97⋅15.6757≈1521 trees.

STEP 13

B(100)=117(1.021)100≈117⋅8.5772≈1003B(100) = 117(1.021)^{100} \approx 117 \cdot 8.5772 \approx \textbf{1003}B(100)=117(1.021)100≈117⋅8.5772≈1003 trees.

STEP 14

Forest A will have more trees after 100 years.The difference is approximately 1521−1003=5181521 - 1003 = \textbf{518}1521−1003=518 more trees.

STEP 15

Forest A is growing faster at a rate of **2.8%** compared to Forest B's **2.1%**.Forest B initially had **20** more trees.After 20 years, Forest B will have approximately **8** more trees.After 100 years, Forest A will have approximately **518** more trees.

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1. Identify Growth Rates
2. Find Initial Populations
3. Calculate Population after 20 Years
4. Calculate Population after 100 Years

Forest B's population function is $B(t) = 117(1.021)^t$ .
The growth factor is $1.021$ .
This means the population increases by $1.021 - 1 = 0.021$ each year.
As a percentage, this is 0.021 \cdot 100\% = \textbf{2.1%} growth per year.

Since $2.8\% > 2.1\%$ , Forest A's population is growing at a faster rate.

The initial population is when $t = 0$ .
So, for Forest A, $A(0) = 97(1.028)^0 = 97 \cdot 1 = \textbf{97}$ trees.

For Forest B, $B(0) = 117(1.021)^0 = 117 \cdot 1 = \textbf{117}$ trees.

Forest B initially had more trees.
The difference is $117 - 97 = \textbf{20}$ more trees.

$A(20) = 97(1.028)^{20} \approx 97 \cdot 1.7535 \approx \textbf{170}$ trees.

$B(20) = 117(1.021)^{20} \approx 117 \cdot 1.5223 \approx \textbf{178}$ trees.

Forest B will have more trees after 20 years.
The difference is approximately $178 - 170 = \textbf{8}$ more trees.

$A(100) = 97(1.028)^{100} \approx 97 \cdot 15.6757 \approx \textbf{1521}$ trees.

$B(100) = 117(1.021)^{100} \approx 117 \cdot 8.5772 \approx \textbf{1003}$ trees.

Forest A will have more trees after 100 years.
The difference is approximately $1521 - 1003 = \textbf{518}$ more trees.

Forest A is growing faster at a rate of 2.8% compared to Forest B's 2.1%.
Forest B initially had 20 more trees.
After 20 years, Forest B will have approximately 8 more trees.
After 100 years, Forest A will have approximately 518 more trees.