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 algae in a test tube is modeled by a logistic differential equation. The maximum capacity of the tube is 580 algae. At 8AM, the number of algae in the test tube is 155 and is increasing at a rate of 31 algae per minute. Write a differential equation to describe the situation. (dP)/(dt)=◻

The rate of change dPdt \frac{d P}{d t} of the number of algae in a test tube is modeled by a logistic differential equation. The maximum capacity of the tube is 580580 algae. At 8AM 8 \mathrm{AM} , the number of algae in the test tube is 155155 and is increasing at a rate of 3131 algae per minute. Write a differential equation to describe the situation.\newlinedPdt=□ \frac{d P}{d t}=\square

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Grade 7right chevron icon
Evaluate two-variable equations: word problemsright chevron icon

Full solution

Q. The rate of change dPdt \frac{d P}{d t} of the number of algae in a test tube is modeled by a logistic differential equation. The maximum capacity of the tube is 580580 algae. At 8AM 8 \mathrm{AM} , the number of algae in the test tube is 155155 and is increasing at a rate of 3131 algae per minute. Write a differential equation to describe the situation.\newlinedPdt=□ \frac{d P}{d t}=\square
  1. Logistic Growth Model Equation: The logistic growth model is given by the differential equation:\newlinedPdt=rP(1−PK)\frac{dP}{dt} = rP(1 - \frac{P}{K})\newlinewhere:\newline- PP is the population at time tt,\newline- rr is the growth rate,\newline- KK is the carrying capacity of the environment (maximum population size).\newlineWe need to find the value of rr and use the given KK value to write the differential equation.
  2. Given Carrying Capacity: We are given the carrying capacity KK as 580580 algae. This value will be used in our differential equation.
  3. Initial Algae Population Data: We are given that at 8AM8\text{AM}, the number of algae is 155155 and is increasing at a rate of 3131 algae per minute. This rate of increase is the value of (dP)/(dt)(dP)/(dt) when P=155P = 155.
  4. Calculate Growth Rate: To find the growth rate rr, we can use the given rate of change when P=155P = 155. Plugging these values into the logistic growth model, we get:\newline31=r×155×(1−155/580)31 = r \times 155 \times (1 - 155/580)\newlineNow we need to solve for rr.
  5. Simplify Fraction: First, simplify the fraction inside the parentheses: 1−155580=1−0.26724≈0.732761 - \frac{155}{580} = 1 - 0.26724 \approx 0.73276
  6. Solve for Growth Rate: Now, plug this value back into the equation and solve for rr:31=r×155×0.7327631 = r \times 155 \times 0.73276r≈31/(155×0.73276)r \approx 31 / (155 \times 0.73276)r≈31/113.5778r \approx 31 / 113.5778r≈0.273r \approx 0.273
  7. Write Differential Equation: Now that we have the value of rr, we can write the logistic differential equation as:\newlinedPdt=0.273P(1−P580)\frac{dP}{dt} = 0.273P(1 - \frac{P}{580})\newlineThis equation models the rate of change of the number of algae in the test tube.

More problems from Evaluate two-variable equations: word problems

Question
This equation shows how the distance traveled by a car is related to the time spent driving. d=50t+10 d = 50t + 10 . The variable t t represents the time in hours, and the variable d d represents the distance in miles. How far does the car travel in 2 2 hours?
Get tutor helpright-arrow

Posted 9 months ago

Question
The equation shows how the cost of buying pencils is related to the number of pencils purchased. \newline c=0.5p+2 c= 0.5p+2 \newlineThe variable p p represents the number of pencils, and the variable c c represents the total cost. How much does it cost to buy 10 10 pencils?
Get tutor helpright-arrow

Posted 9 months ago

Question
This equation shows how the total score in a game is related to the number of points scored per level. s=100l+250 s = 100l + 250 . The variable l l represents the levels completed, and the variable s s represents the total score. What is the total score after completing 3 3 levels?
Get tutor helpright-arrow

Posted 7 months ago

Question
The equation shows how the amount of paint needed is related to the area to be painted. p=2a+5 p = 2a + 5 . The variable a a represents the area in square meters, and the variable p p represents the amount of paint in liters. How much paint is needed for an area of 8 8 square meters?
Get tutor helpright-arrow

Posted 7 months ago

Question
This equation shows how the total bill for a meal is related to the number of people sharing the meal. \newline b=15n+35 b = 15n + 35 \newlineThe variable n n represents the number of people, and the variable b b represents the total bill. What is the total bill for a meal shared by 4 4 people?
Get tutor helpright-arrow

Posted 9 months ago

Question
Poppy the dog weighs p p pounds, and Yumi the cat weighs y y pounds. If Poppy weighs 3636 more pounds than Yumi does, which of the following equations correctly describes the relationship between their weights?\newlineChoose 11 answer:\newline(A) p+y=36 p+y=36 \newline(B) p+36=y p+36=y \newline(C) p−y=36 p-y=36 \newline(D) y−p=36 y-p=36
Get tutor helpright-arrow

Posted 7 months ago

Question
Calvin and Melvin are two of Santa's elves. The number of toys they build is given by 11c+8m 11 c+8 m , where c c is the number of candy canes Calvin eats for breakfast and m m is the number of candy canes Melvin eats for breakfast. How many toys do they build when Calvin eats 55 candy canes and Melvin eats 33 candy canes for breakfast?
Get tutor helpright-arrow

Posted 7 months ago

Question
2w+4 2 w+4 \newlineCarly is cutting a piece of paper stock based on the width of a photo. The expression above describes the length of the paper stock, in inches, for a photo that is w w inches wide. If the photo is 88.55 inches wide, what is the length of the paper stock in inches?
Get tutor helpright-arrow

Posted 7 months ago

Question
Calvin and Melvin are two of Santa's elves. The number of toys they build is given by 11c+8m 11 c+8 m where c c is the number of candy canes Calvin eats for breakfast and m m is the number of candy canes Melvin eats for breakfast.\newlineHow many toys do they build when Calvin eats 55 candy canes and Melvin eats 33 candy canes for breakfast?\newlinetoys
Get tutor helpright-arrow

Posted 8 months ago

Question
The approval rating of a certain politician is changing at a rate of r(t) r(t) percent of the population per month (where t t is the time in months).\newlineWhat does ∫03r(t)dt \int_{0}^{3} r(t) d t represent?\newlineChoose 11 answer:\newline(A) The rate of change of the approval rating of the politician when t=3 t=3 months\newline(B) The average rate of change of the approval rating during the first three months\newline(C) The change in the percent of the population who approves of the politician during the first three months\newline(D) The percent of the population who approves of the politician when t=3 t=3 months
Get tutor helpright-arrow

Posted 7 months ago

View all (180)

Related topics

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