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Chapter 10: Problem 37

Suppose that, once a sunflower plant has started growing, the rate of growth at any time is proportional to the product of its height and the difference between its height at maturity and its current height. Give a differential equation that is satisfied by \(f(t),\) the height at time \(t,\) and sketch the solution.

Short Answer

Expert verified
The differential equation is \[ \frac{df(t)}{dt} = k f(t) (H - f(t)) \]. The solution is a sigmoidal curve.

Step by step solution

01

Identify given information

The rate of growth of the sunflower is proportional to the product of its height and the difference between its height at maturity and its current height. Let the height at time t be denoted by f(t) and the height at maturity be H.
02

Set up the proportional relationship

According to the problem, the growth rate is proportional to the product of the height and the difference between H and the current height. Therefore, the rate of growth can be expressed as: \( \frac{df(t)}{dt} = k f(t) (H - f(t)) \) where k is the proportionality constant.
03

Write the differential equation

Combining all the information, we get the following differential equation: \[ \frac{df(t)}{dt} = k f(t) (H - f(t)) \] where f(t) is the height at time t, H is the height at maturity, and k is the proportionality constant.
04

Sketch the solution

Given the initial condition where the height starts from a value much smaller than H, the function f(t) will follow a sigmoidal (S-shaped) curve. It will grow slowly at first, then rapidly when close to half of the maturity height, and finally slow down as it approaches the height at maturity H.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Growth Rate Modeling
In biology, growth rate modeling involves understanding how organisms increase in size over time. Growth rate is often affected by various factors, such as availability of nutrients, environmental conditions, and genetic traits.

For the sunflower plant described in the exercise, the growth rate can be modeled using a differential equation. The sunflower's height at time \(t\), denoted by \(f(t)\), grows in proportion to its current height and the remaining growth potential towards maturity. This considers the effect of both current size and remaining capacity for growth.

We can use this model to predict how the sunflower will grow over time, allowing us to study and understand the underlying biological processes more effectively.
Proportional Relationships
A proportional relationship indicates that one quantity changes at a rate that depends on another. In our exercise, the growth rate of the sunflower is proportional to the product of its current height and the difference between its mature height and current height.

This can be mathematically expressed as:
\frac{df(t)}{dt} = k f(t) (H - f(t))
Here:

  • \frac{df(t)}{dt} is the rate of change of height (\(f(t)\)) over time (\(t\))
  • \(k\) is the proportionality constant
  • \(H\) is the mature height

Understanding this proportional relationship helps us capture the growth dynamics of the sunflower plant, demonstrating how biological processes often depend on interactions between different factors.
Differential Equations Structure
Differential equations are mathematical equations that involve functions and their derivatives. They are used to describe various phenomena, including growth processes in biology.

The differential equation in our exercise is:
\frac{df(t)}{dt} = k f(t) (H - f(t))
This equation reflects the dynamic nature of the sunflower's growth. Each component of the equation has a specific meaning:
  • \frac{df(t)}{dt} represents the rate of growth of the sunflower
  • \(f(t)\) represents the current height at time \(t\)
  • \((H - f(t))\) indicates the remaining growth to reach maturity
  • \(k\) is a constant that scales the proportionality

Analyzing this structure allows us to understand how each factor influences the growth rate, helping to visualize and predict the behavior of biological systems over time.

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Most popular questions from this chapter

Suppose that substance \(A\) is converted into substance \(B\) at a rate that, at any time \(t,\) is proportional to the square of the amount of \(A\). This situation occurs, for instance, when it is necessary for two molecules of \(A\) to collide to create one molecule of \(B\). Set up the differential equation that is satisfied by \(y=f(t),\) the amount of substance \(A\) at time \(t .\) Sketch a solution.

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