Math / Algebra

QuestionA certain forest covers an area of $4000 \mathrm{~km}^{2}$ . Suppose that each year this area decreases by $3 \%$ . What will the area be after 14 years? Use the calculator provided and round your answer to the nearest square kilometer. $\square$ $m^{2}$

Studdy Solution

STEP 1

1. The initial area of the forest is $4000 \, \text{km}^2$ .
2. The area decreases by $3\%$ each year.
3. We are calculating the area after 14 years.
4. The answer should be rounded to the nearest square kilometer.

STEP 2

1. Understand the concept of exponential decay.
2. Define the formula for exponential decay.
3. Substitute the given values into the formula.
4. Calculate the area after 14 years.
5. Round the result to the nearest square kilometer.

STEP 3

Understand the concept of exponential decay. The area decreases by a fixed percentage each year, which is modeled by exponential decay.

STEP 4

Define the formula for exponential decay. The formula for exponential decay is:
$A = A_0 \times (1 - r)^t$
where: - $A$ is the final amount (area after 14 years), - $A_0$ is the initial amount (initial area), - $r$ is the decay rate (as a decimal), - $t$ is the time in years.

STEP 5

Substitute the given values into the formula. Here, $A_0 = 4000 \, \text{km}^2$ , $r = 0.03$ , and $t = 14$ .
$A = 4000 \times (1 - 0.03)^{14}$

STEP 6

Calculate the area after 14 years using the provided calculator.
First, calculate $(1 - 0.03)^{14}$ :
$(1 - 0.03) = 0.97$
$0.97^{14} \approx 0.642$
Now, calculate the final area:
$A = 4000 \times 0.642 \approx 2568$

STEP 7

Round the result to the nearest square kilometer. The area after 14 years is approximately:
$\boxed{2568} \, \text{km}^2$

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QuestionA certain forest covers an area of $4000 \mathrm{~km}^{2}$ . Suppose that each year this area decreases by $3 \%$ . What will the area be after 14 years? Use the calculator provided and round your answer to the nearest square kilometer. $\square$ $m^{2}$

Studdy Solution

STEP 1

1. The initial area of the forest is $4000 \, \text{km}^2$ .
2. The area decreases by $3\%$ each year.
3. We are calculating the area after 14 years.
4. The answer should be rounded to the nearest square kilometer.

STEP 2

1. Understand the concept of exponential decay.
2. Define the formula for exponential decay.
3. Substitute the given values into the formula.
4. Calculate the area after 14 years.
5. Round the result to the nearest square kilometer.

STEP 3

Understand the concept of exponential decay. The area decreases by a fixed percentage each year, which is modeled by exponential decay.

STEP 4

Define the formula for exponential decay. The formula for exponential decay is:
$A = A_0 \times (1 - r)^t$
where: - $A$ is the final amount (area after 14 years), - $A_0$ is the initial amount (initial area), - $r$ is the decay rate (as a decimal), - $t$ is the time in years.

STEP 5

Substitute the given values into the formula. Here, $A_0 = 4000 \, \text{km}^2$ , $r = 0.03$ , and $t = 14$ .
$A = 4000 \times (1 - 0.03)^{14}$

STEP 6

Calculate the area after 14 years using the provided calculator.
First, calculate $(1 - 0.03)^{14}$ :
$(1 - 0.03) = 0.97$
$0.97^{14} \approx 0.642$
Now, calculate the final area:
$A = 4000 \times 0.642 \approx 2568$

STEP 7

Round the result to the nearest square kilometer. The area after 14 years is approximately:
$\boxed{2568} \, \text{km}^2$

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QuestionA certain forest covers an area of 4000 km24000 \mathrm{~km}^{2}4000 km2. Suppose that each year this area decreases by 3%3 \%3%. What will the area be after 14 years? Use the calculator provided and round your answer to the nearest square kilometer. □\square□ m2m^{2}m2

Studdy Solution

STEP 1

1. The initial area of the forest is 4000 km2 4000 \, \text{km}^2 4000km2.2. The area decreases by 3% 3\% 3% each year.3. We are calculating the area after 14 years.4. The answer should be rounded to the nearest square kilometer.

STEP 2

1. Understand the concept of exponential decay.2. Define the formula for exponential decay.3. Substitute the given values into the formula.4. Calculate the area after 14 years.5. Round the result to the nearest square kilometer.

STEP 3

Understand the concept of exponential decay. The area decreases by a fixed percentage each year, which is modeled by exponential decay.

STEP 4

STEP 5

Substitute the given values into the formula. Here, A0=4000 km2 A_0 = 4000 \, \text{km}^2 A0​=4000km2, r=0.03 r = 0.03 r=0.03, and t=14 t = 14 t=14.A=4000×(1−0.03)14 A = 4000 \times (1 - 0.03)^{14} A=4000×(1−0.03)14

STEP 6

STEP 7

Round the result to the nearest square kilometer. The area after 14 years is approximately:2568 km2 \boxed{2568} \, \text{km}^2 2568​km2

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QuestionA certain forest covers an area of $4000 \mathrm{~km}^{2}$ . Suppose that each year this area decreases by $3 \%$ . What will the area be after 14 years? Use the calculator provided and round your answer to the nearest square kilometer. $\square$ $m^{2}$

1. The initial area of the forest is $4000 \, \text{km}^2$ .
2. The area decreases by $3\%$ each year.
3. We are calculating the area after 14 years.
4. The answer should be rounded to the nearest square kilometer.

1. Understand the concept of exponential decay.
2. Define the formula for exponential decay.
3. Substitute the given values into the formula.
4. Calculate the area after 14 years.
5. Round the result to the nearest square kilometer.

Substitute the given values into the formula. Here, $A_0 = 4000 \, \text{km}^2$ , $r = 0.03$ , and $t = 14$ .
$A = 4000 \times (1 - 0.03)^{14}$

Round the result to the nearest square kilometer. The area after 14 years is approximately:
$\boxed{2568} \, \text{km}^2$