Math / Algebra

QuestionUse the approximate half-life formula for the case described below. Discuss whether the formula is valid for the case described. Poaching is causing a population of elephants to decline by $8 \%$ per year. What is the half-life for the population? If there are 10,000 elephants today, how many will remain in 60 years?
Use the approximate half-life formula. What is the half-life for the population? $\square$ years (Type an integer or decimal rounded to the nearest hundredth as needed.)

Studdy Solution

STEP 1

What is this asking? If elephants are decreasing by 8% yearly, how long until half are left, and how many will remain after 60 years if we start with 10,000? Watch out! Don't mix up the rate of decline with the remaining population.
Also, make sure to use the correct half-life formula!

STEP 2

1. Calculate the half-life.
2. Calculate the remaining population after 60 years.

STEP 3

Alright class, let's calculate the half-life!
The approximate half-life formula is a great tool for estimating how long it takes for a quantity to decrease by half.
It's super useful in situations like this where we're dealing with exponential decay.

STEP 4

The formula is $T_{half} \approx \frac{70}{P}$ , where $P$ is the percentage of decrease per year.
In our case, the elephant population is declining by 8% per year, so $P = 8$ .

STEP 5

Let's plug that into our formula: $T_{half} \approx \frac{70}{8}$ .

STEP 6

Now, let's crunch those numbers: $T_{half} \approx 8.75$ .
So, the half-life of the elephant population is approximately 8.75 years.
That means, if this trend continues, every 8.75 years the number of elephants will be cut in half!

STEP 7

Now, let's figure out how many elephants will be left after 60 years.
We'll use the formula for exponential decay: $A = A_0 \cdot (1 - r)^t$ , where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the rate of decay (as a decimal), and $t$ is the time in years.

STEP 8

We know that the initial population $A_0$ is 10,000, the rate of decay $r$ is 0.08 (8% expressed as a decimal), and the time $t$ is 60 years.
Let's plug these values into our formula: $A = 10000 \cdot (1 - 0.08)^{60}$ .

STEP 9

First, simplify inside the parentheses: $A = 10000 \cdot (0.92)^{60}$ .

STEP 10

Now, we raise 0.92 to the power of 60: $A = 10000 \cdot 0.007847$ (approximately).

STEP 11

Finally, multiply by 10,000: $A \approx 78.47$ .
Since we're talking about elephants, we can't have a fraction of an elephant, so we'll round down to the nearest whole number.

STEP 12

The half-life of the elephant population is approximately 8.75 years.
In 60 years, approximately 78 elephants will remain.

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Studdy Solution

STEP 1

What is this asking? If elephants are decreasing by 8% yearly, how long until half are left, and how many will remain after 60 years if we start with 10,000? Watch out! Don't mix up the rate of decline with the remaining population.
Also, make sure to use the correct half-life formula!

STEP 2

1. Calculate the half-life.
2. Calculate the remaining population after 60 years.

STEP 3

Alright class, let's calculate the half-life!
The approximate half-life formula is a great tool for estimating how long it takes for a quantity to decrease by half.
It's super useful in situations like this where we're dealing with exponential decay.

STEP 4

The formula is $T_{half} \approx \frac{70}{P}$ , where $P$ is the percentage of decrease per year.
In our case, the elephant population is declining by 8% per year, so $P = 8$ .

STEP 5

Let's plug that into our formula: $T_{half} \approx \frac{70}{8}$ .

STEP 6

Now, let's crunch those numbers: $T_{half} \approx 8.75$ .
So, the half-life of the elephant population is approximately 8.75 years.
That means, if this trend continues, every 8.75 years the number of elephants will be cut in half!

STEP 7

Now, let's figure out how many elephants will be left after 60 years.
We'll use the formula for exponential decay: $A = A_0 \cdot (1 - r)^t$ , where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the rate of decay (as a decimal), and $t$ is the time in years.

STEP 8

We know that the initial population $A_0$ is 10,000, the rate of decay $r$ is 0.08 (8% expressed as a decimal), and the time $t$ is 60 years.
Let's plug these values into our formula: $A = 10000 \cdot (1 - 0.08)^{60}$ .

STEP 9

First, simplify inside the parentheses: $A = 10000 \cdot (0.92)^{60}$ .

STEP 10

Now, we raise 0.92 to the power of 60: $A = 10000 \cdot 0.007847$ (approximately).

STEP 11

Finally, multiply by 10,000: $A \approx 78.47$ .
Since we're talking about elephants, we can't have a fraction of an elephant, so we'll round down to the nearest whole number.

STEP 12

The half-life of the elephant population is approximately 8.75 years.
In 60 years, approximately 78 elephants will remain.

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Studdy Solution

STEP 1

What is this asking? If elephants are decreasing by 8% yearly, how long until half are left, and how many will remain after 60 years if we start with 10,000? Watch out! Don't mix up the *rate of decline* with the *remaining population*.Also, make sure to use the correct half-life formula!

STEP 2

1. Calculate the half-life.2. Calculate the remaining population after 60 years.

STEP 3

Alright class, let's **calculate the half-life**!The approximate half-life formula is a great tool for estimating how long it takes for a quantity to decrease by half.It's super useful in situations like this where we're dealing with exponential decay.

STEP 4

The formula is Thalf≈70P T_{half} \approx \frac{70}{P} Thalf​≈P70​, where PPP is the **percentage of decrease** per year.In our case, the elephant population is declining by **8%** per year, so P=8 P = 8 P=8.

STEP 5

Let's plug that into our formula: Thalf≈708 T_{half} \approx \frac{70}{8} Thalf​≈870​.

STEP 6

Now, let's **crunch those numbers**: Thalf≈8.75 T_{half} \approx 8.75 Thalf​≈8.75.So, the half-life of the elephant population is approximately **8.75 years**.That means, if this trend continues, every 8.75 years the number of elephants will be cut in half!

STEP 7

STEP 8

We know that the **initial population** A0A_0A0​ is **10,000**, the **rate of decay** rrr is **0.08** (8% expressed as a decimal), and the **time** ttt is **60 years**.Let's plug these values into our formula: A=10000⋅(1−0.08)60 A = 10000 \cdot (1 - 0.08)^{60} A=10000⋅(1−0.08)60.

STEP 9

First, simplify inside the parentheses: A=10000⋅(0.92)60 A = 10000 \cdot (0.92)^{60} A=10000⋅(0.92)60.

STEP 10

Now, we raise 0.92 to the power of 60: A=10000⋅0.007847 A = 10000 \cdot 0.007847 A=10000⋅0.007847 (approximately).

STEP 11

Finally, multiply by 10,000: A≈78.47 A \approx 78.47 A≈78.47.Since we're talking about elephants, we can't have a fraction of an elephant, so we'll round down to the nearest whole number.

STEP 12

The half-life of the elephant population is approximately **8.75 years**.In 60 years, approximately **78** elephants will remain.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? If elephants are decreasing by 8% yearly, how long until half are left, and how many will remain after 60 years if we start with 10,000? Watch out! Don't mix up the rate of decline with the remaining population.
Also, make sure to use the correct half-life formula!

1. Calculate the half-life.
2. Calculate the remaining population after 60 years.

Alright class, let's calculate the half-life!
The approximate half-life formula is a great tool for estimating how long it takes for a quantity to decrease by half.
It's super useful in situations like this where we're dealing with exponential decay.

The formula is $T_{half} \approx \frac{70}{P}$ , where $P$ is the percentage of decrease per year.
In our case, the elephant population is declining by 8% per year, so $P = 8$ .

Let's plug that into our formula: $T_{half} \approx \frac{70}{8}$ .

Now, let's crunch those numbers: $T_{half} \approx 8.75$ .
So, the half-life of the elephant population is approximately 8.75 years.
That means, if this trend continues, every 8.75 years the number of elephants will be cut in half!

We know that the initial population $A_0$ is 10,000, the rate of decay $r$ is 0.08 (8% expressed as a decimal), and the time $t$ is 60 years.
Let's plug these values into our formula: $A = 10000 \cdot (1 - 0.08)^{60}$ .

First, simplify inside the parentheses: $A = 10000 \cdot (0.92)^{60}$ .

Now, we raise 0.92 to the power of 60: $A = 10000 \cdot 0.007847$ (approximately).

Finally, multiply by 10,000: $A \approx 78.47$ .
Since we're talking about elephants, we can't have a fraction of an elephant, so we'll round down to the nearest whole number.

The half-life of the elephant population is approximately 8.75 years.
In 60 years, approximately 78 elephants will remain.