Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The rate of change (dP)/(dt) of the number of people on a beach is modeled by the following differential equation: (dP)/(dt)=(6154)/(21945)P(1-(P)/( 724)) At t=0, the number of people on the beach is 154 and is increasing at a rate of 34 people per hour. Find lim_(t rarr oo)P(t). Answer:

The rate of change dPdt \frac{d P}{d t} of the number of people on a beach is modeled by the following differential equation:\newlinedPdt=615421945P(1−P724) \frac{d P}{d t}=\frac{6154}{21945} P\left(1-\frac{P}{724}\right) \newlineAt t=0 t=0 , the number of people on the beach is 154154 and is increasing at a rate of 3434 people per hour. Find lim⁡t→∞P(t) \lim _{t \rightarrow \infty} P(t) .\newlineAnswer:

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Write and solve direct variation equationsright chevron icon

Full solution

Q. The rate of change dPdt \frac{d P}{d t} of the number of people on a beach is modeled by the following differential equation:\newlinedPdt=615421945P(1−P724) \frac{d P}{d t}=\frac{6154}{21945} P\left(1-\frac{P}{724}\right) \newlineAt t=0 t=0 , the number of people on the beach is 154154 and is increasing at a rate of 3434 people per hour. Find lim⁡t→∞P(t) \lim _{t \rightarrow \infty} P(t) .\newlineAnswer:
  1. Identify Equation and Equilibrium: Identify the differential equation and the equilibrium point.\newlineThe differential equation given is dPdt=615421945P(1−P724)\frac{dP}{dt} = \frac{6154}{21945}P\left(1 - \frac{P}{724}\right). This is a logistic growth model where the growth rate of the population PP is proportional to both the current population and the room left for growth, which is (1−P724)\left(1 - \frac{P}{724}\right). The equilibrium point is where the growth rate dPdt\frac{dP}{dt} becomes zero, which happens when P=724P = 724, since 1−P724=01 - \frac{P}{724} = 0 at that point.
  2. Population Behavior at Infinity: Determine the behavior of the population as tt approaches infinity. In logistic growth models, the population tends to stabilize at the equilibrium point as tt approaches infinity. This means that the limit of P(t)P(t) as tt approaches infinity should be the equilibrium population, which is P=724P = 724.
  3. Verify Initial Conditions: Verify the initial conditions and the rate of change.\newlineAt t=0t=0, we are given that P(0)=154P(0) = 154 and dPdt=34\frac{dP}{dt} = 34 people per hour. This information is consistent with the model because the initial rate of change is positive, indicating that the population is increasing towards the equilibrium.
  4. Conclude Population Limit: Conclude the limit of P(t)P(t) as tt approaches infinity. Since the differential equation models logistic growth and we have determined that the equilibrium point is P=724P = 724, the limit of P(t)P(t) as tt approaches infinity is 724724.

More problems from Write and solve direct variation equations

Question
In a direct variation, y=6 y = 6 when x=2 x = 2 . Write a direct variation equation that shows the relationship between x x and y y .\newlineWrite your answer as an equation with y y first, followed by an equals sign.
Get tutor helpright-arrow

Posted 5 months ago

Question
If y y varies directly with x x and y=12 y = 12 when x=6 x = 6 , find y y when x=4 x = 4 .\newlineWrite and solve a direct variation equation to find the answer.\newliney= y = ____
Get tutor helpright-arrow

Posted 5 months ago

Question
Which equation shows inverse variation?\newlineChoices:\newline(x36)=y(\frac{x}{36}) = y\newlineyx=−74yx = -74
Get tutor helpright-arrow

Posted 5 months ago

Question
Assume that y y varies inversely with x x . If y=1 y = 1 when x=4 x = 4 , find y y when x=2 x = 2 .\newlineWrite and solve an inverse variation equation to find the answer.\newliney= y = _____
Get tutor helpright-arrow

Posted 5 months ago

Question
What is the value of x2y4 \frac{x^{2}}{y^{4}} when x=8 x=8 and y=2 y=2 ?
Get tutor helpright-arrow

Posted 5 months ago

Question
(x+4)(2x−a)=0 (x+4)(2 x-a)=0 \newlineIn the given equation, a a is a constant. If the equation has the solutions x=4 x=4 and x=−4 x=-4 , what is the value of a a ?
Get tutor helpright-arrow

Posted 3 months ago

Question
(2x+5)(−mx+9)=0 (2 x+5)(-m x+9)=0 \newlineIn the given equation, m m is a constant. If the equation has the solutions x=−52 x=-\frac{5}{2} and x=32 x=\frac{3}{2} , what is the value of m m ?
Get tutor helpright-arrow

Posted 3 months ago

Question
(4x+3)−(5x−7)=ax+b (4 x+3)-(5 x-7)=a x+b \newlineIn the given equation, b b is a constant. What is the value of b b ?
Get tutor helpright-arrow

Posted 3 months ago

Question
y=298x−23 y=\frac{29}{8} x-\frac{2}{3} \newlineThe given equation is graphed in the xy x y -plane. What is the slope of the line?
Get tutor helpright-arrow

Posted 8 months ago

Question
at2+72t−4=0 a t^{2}+\frac{7}{2} t-4=0 \newlineThe given equation has solutions at t=−8 t=-8 and t=1 t=1 . What is the value of the constant a a ?
Get tutor helpright-arrow

Posted 8 months ago

View all (80)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant