1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
alexdok [17]
3 years ago
5

The logistic equation for the population​ (in thousands) of a certain species is given by:

Mathematics
1 answer:
Eva8 [605]3 years ago
6 0

Answer:

a.

b. 1.5

c. 1.5

d. No

Step-by-step explanation:

a. First, let's solve the differential equation:

\frac{dp}{dt} =3p-2p^2

Divide both sides by 3p-2p^2  and multiply both sides by dt:

\frac{dp}{3p-2p^2}=dt

Integrate both sides:

\int\ \frac{1}{3p-2p^2}  dp =\int\ dt

Evaluate the integrals and simplify:

p(t)=\frac{3e^{3t} }{C_1+2e^{3t}}

Where C1 is an arbitrary constant

I sketched the direction field using a computer software. You can see it in the picture that I attached you.

b. First let's find the constant C1 for the initial condition given:

p(0)=3=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}

Solving for C1:

C_1=-1

Now, let's evaluate the limit:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-1 }  \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2-e^{-3x}  }

The expression -e^{-3x} tends to zero as x approaches ∞ . Hence:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-1 } =\frac{3}{2} =1.5

c. As we did before, let's find the constant C1 for the initial condition given:

p(0)=0.8=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}

Solving for C1:

C_1=1.75

Now, let's evaluate the limit:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}+1.75 }  \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2+1.75e^{-3x}  }

The expression -e^{-3x} tends to zero as x approaches ∞ . Hence:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}+1.75 } =\frac{3}{2} =1.5

d. To figure out that, we need to do the same procedure as we did before. So,  let's find the constant C1 for the initial condition given:

p(0)=2=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}

Solving for C1:

C_1=-\frac{1}{2} =-0.5

Can a population of 2000 ever decline to 800? well, let's find the limit of the function when it approaches to ∞:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-0.5 }  \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2-0.5e^{-3x}  }

The expression -e^{-3x} tends to zero as x approaches ∞ . Hence:

\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-0.5 } =\frac{3}{2} =1.5

Therefore, a population of 2000 never will decline to 800.

You might be interested in
Solve pls brainliest
gizmo_the_mogwai [7]

Answer:

5/6

Step-by-step explanation:

11/12 - 1/12 = 10/12

10 divided by 2 = 5
12 divided by 2 = 6

5/6

8 0
2 years ago
Read 2 more answers
The radius r of a circle is increasing at a rate of 8 centimeters per minute. find the rate of change of the area when r = 39 ce
Vitek1552 [10]
Dr/dt= 8cmM-¹
dA/dt = ?
but A= πr²
so dA/dr =2πr
but recall
dA/dt = dA/dr • dr/dt
dA/dt = 2πr • 8cmM-¹
dA/dt = 2×π×39 ×8 = 624cm²M-1
7 0
3 years ago
amir lent Rs. 600 to hameed for 2 years and Rs. 150 to aman for 4 year and received altogether from both Rs. 90 as simple intere
aniked [119]

Interest = Pin = principal * interest rate * number of years

Let interest rate be i, then
600*i*2+150*i*4=90
1200i+600i = 90
i=90/(1200+600)=90/1800=1/20=0.05 or 5%

8 0
3 years ago
Below are two parallel lines with a third line intersecting them.
Allushta [10]
180-48= 132 ( I think)
8 0
2 years ago
What is m∠G ?<br><br> Enter your answer in the box.
Elena-2011 [213]
We are given all the angle measures, therefore we can equate all the expressions to 180 degrees. We can do this by using the Triangle Angle Sum Theorem, which states that all triangles' interior angles add up to 180 degrees. After we do that, we get:
x + 3x + 25 + x - 5 = 180
Now, just solve for x:
x + 3x + 25 + x - 5 = 180
5x + 20 = 180
5x = 160
x = 32
We've found the value of x, but the question asks us the measure of angle G. We can find the angle measure by substituting 32 into the expression for angle G. Doing so and simplifying gives us 121 degrees. Therefore, the measure of angle G is 121 degrees. Hope my answer has come of aid to you and have a great day!
6 0
3 years ago
Read 2 more answers
Other questions:
  • Leona spent $22.95 at the mall. She bought a T-shirt and a book. The book $12.25. How much did the T-shirt cost?
    15·1 answer
  • The answer and how to solve pls
    9·1 answer
  • A walking path across a park is represented by the equation A walking path across a park is represented by the equation y= -3x-3
    6·1 answer
  • What is the opssit of 5.5​
    7·1 answer
  • The coordinate for the Orgin in the coordinate plane are ______
    11·1 answer
  • Sample Response/Explanation: No. The correct values of a, b, and c were substituted in, but the formula was simplified wrong. Th
    14·2 answers
  • The number 4 + i is multiplied by some real number to yield 2 + one-half i. Which equation is true?
    7·2 answers
  • Can someone please help me???
    8·2 answers
  • Which graph represents the solution set for the inequality 7.5 &lt; x?
    8·1 answer
  • What is the linear function equation represented by the graph? Enter your answer in the box. f(x) =​
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!