Math / Calculus

QuestionQuestion 3 Suppose that there are initially 600 rabbits in a population, and that the population changes at a rate of $f(t)=t$ where $t$ is in weeks. How many rabbits will there be after 4 weeks?

Studdy Solution

STEP 1

What is this asking? We've got a bunch of bunnies, and their numbers are growing each week, and we want to know how many there'll be after a month! Watch out! The rate of change isn't constant, it's increasing each week!

STEP 2

1. Find the change in rabbit population.
2. Calculate the final rabbit population.

STEP 3

The problem tells us that the population changes at a rate of $f(t) = t$ rabbits per week, where $t$ is the number of weeks.
This means that after one week, the population increases by $f(1) = 1$ rabbit.
After two weeks, it increases by $f(2) = 2$ rabbits per week, and so on.
To find the total change in population over the four weeks, we need to add up the changes for each week.

STEP 4

Since the rate of change is given by $f(t) = t$ , we can think of the total change as the area under the graph of this function from $t = 0$ to $t = 4$ .
This is given by the definite integral: $\int_0^4 t \, dt$

STEP 5

Let's calculate the integral: $\int_0^4 t \, dt = \left[ \frac{1}{2} t^2 \right]_0^4$ Remember, we add one to the exponent, which makes it $t^2$ , and then divide by the new exponent, which gives us $\frac{1}{2}t^2$ .

STEP 6

Now we evaluate the integral at the upper and lower limits of integration: $\left[ \frac{1}{2} t^2 \right]_0^4 = \left( \frac{1}{2} \cdot 4^2 \right) - \left( \frac{1}{2} \cdot 0^2 \right) = \frac{1}{2} \cdot 16 - 0 = 8$ So, the total change in the rabbit population after 4 weeks is 8 rabbits.

STEP 7

We started with an initial population of $600$ rabbits.
We just calculated that the population increased by 8 rabbits over four weeks.

STEP 8

To find the final population, we simply add the change in population to the initial population: $600 + 8 = 608$

STEP 9

After 4 weeks, there will be $608$ rabbits!

Was this helpful?

QuestionQuestion 3 Suppose that there are initially 600 rabbits in a population, and that the population changes at a rate of $f(t)=t$ where $t$ is in weeks. How many rabbits will there be after 4 weeks?

Studdy Solution

STEP 1

What is this asking? We've got a bunch of bunnies, and their numbers are growing each week, and we want to know how many there'll be after a month! Watch out! The rate of change isn't constant, it's increasing each week!

STEP 2

1. Find the change in rabbit population.
2. Calculate the final rabbit population.

STEP 3

STEP 4

Since the rate of change is given by $f(t) = t$ , we can think of the total change as the area under the graph of this function from $t = 0$ to $t = 4$ .
This is given by the definite integral: $\int_0^4 t \, dt$

STEP 5

Let's calculate the integral: $\int_0^4 t \, dt = \left[ \frac{1}{2} t^2 \right]_0^4$ Remember, we add one to the exponent, which makes it $t^2$ , and then divide by the new exponent, which gives us $\frac{1}{2}t^2$ .

STEP 6

Now we evaluate the integral at the upper and lower limits of integration: $\left[ \frac{1}{2} t^2 \right]_0^4 = \left( \frac{1}{2} \cdot 4^2 \right) - \left( \frac{1}{2} \cdot 0^2 \right) = \frac{1}{2} \cdot 16 - 0 = 8$ So, the total change in the rabbit population after 4 weeks is 8 rabbits.

STEP 7

We started with an initial population of $600$ rabbits.
We just calculated that the population increased by 8 rabbits over four weeks.

STEP 8

To find the final population, we simply add the change in population to the initial population: $600 + 8 = 608$

STEP 9

After 4 weeks, there will be $608$ rabbits!

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.

QuestionQuestion 3 Suppose that there are initially 600 rabbits in a population, and that the population changes at a rate of f(t)=tf(t)=tf(t)=t where ttt is in weeks. How many rabbits will there be after 4 weeks?

Studdy Solution

STEP 1

What is this asking? We've got a bunch of bunnies, and their numbers are growing each week, and we want to know how many there'll be after a month! Watch out! The rate of change isn't constant, it's increasing each week!

STEP 2

1. Find the change in rabbit population.2. Calculate the final rabbit population.

STEP 3

STEP 4

Since the rate of change is given by f(t)=tf(t) = tf(t)=t, we can think of the total change as the area under the graph of this function from t=0t = 0t=0 to t=4t = 4t=4.This is given by the definite integral: ∫04t dt \int_0^4 t \, dt ∫04​tdt

STEP 5

Let's **calculate the integral**: ∫04t dt=[12t2]04 \int_0^4 t \, dt = \left[ \frac{1}{2} t^2 \right]_0^4 ∫04​tdt=[21​t2]04​ Remember, we add one to the exponent, which makes it t2t^2t2, and then divide by the new exponent, which gives us 12t2\frac{1}{2}t^221​t2.

STEP 6

STEP 7

We started with an **initial population** of 600600600 rabbits.We just calculated that the population *increased* by **8** rabbits over four weeks.

STEP 8

To find the **final population**, we simply add the change in population to the initial population: 600+8=608 600 + 8 = 608 600+8=608

STEP 9

After 4 weeks, there will be 608608608 rabbits!

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.

QuestionQuestion 3 Suppose that there are initially 600 rabbits in a population, and that the population changes at a rate of $f(t)=t$ where $t$ is in weeks. How many rabbits will there be after 4 weeks?

1. Find the change in rabbit population.
2. Calculate the final rabbit population.

Since the rate of change is given by $f(t) = t$ , we can think of the total change as the area under the graph of this function from $t = 0$ to $t = 4$ .
This is given by the definite integral: $\int_0^4 t \, dt$

Let's calculate the integral: $\int_0^4 t \, dt = \left[ \frac{1}{2} t^2 \right]_0^4$ Remember, we add one to the exponent, which makes it $t^2$ , and then divide by the new exponent, which gives us $\frac{1}{2}t^2$ .

We started with an initial population of $600$ rabbits.
We just calculated that the population increased by 8 rabbits over four weeks.

To find the final population, we simply add the change in population to the initial population: $600 + 8 = 608$

After 4 weeks, there will be $608$ rabbits!