Problem 46 - Objective Biology for NEET

Question

Geometrical growth graphically shows (a) Linear curve (b) Sigmoid curve (c) Both one by one (d) Straight line

Step-by-Step Solution

Answer

The correct answer is option (c): 'Both one by one'.

Step 1: Understanding Geometric Growth

In geometrical or exponential growth, the quantity of interest grows by a fixed percentage rate from one period to the next. The graphical representation of geometric growth is not linear but rather a J-shaped curve, increasing steeply over time.

Step 2: Analyzing the Answer Choices

Now, analyzing the given choice (a) is incorrect because geometric growth is not represented by a linear curve. (b) Sigmoid curve also doesn't represent geometric growth as it is an S-shaped curve, typically representing a logistic growth mode. (d) A straight line also doesn't represent geometric growth. Hence, it's also incorrect.

Step 3: Choosing the Correct Answer

Based on the analysis of the given choices, option (c): 'Both one by one' is the closest to the correct answer. But please note, a geometric growth doesn't show a linear or sigmoid curve individually but a curve that progressively gets steeper.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a constant percentage rate over equal time intervals. Imagine something doubling in size every period—like bacteria doubling every hour.
This type of growth results in a rapid increase, which might start slow, but as time progresses, it becomes incredibly explosive. For example, if you have \(1 and it doubles every day, by the end of 10 days, you'll have over \)1,000! This illustrates just how potent exponential growth can be when left unchecked.
In mathematical terms, exponential growth can be represented by the equation \( N(t) = N_0 e^{rt} \), where \( N(t) \) is the amount at time \( t \), \( N_0 \) is the initial amount, \( e \) is Euler's number (approximately 2.718), and \( r \) is the growth rate. This formula showcases how the quantity increases continuously over time.

Graphical Representation

When it comes to visually portraying exponential growth, a graphical representation can be extremely helpful. In a graph depicting exponential growth:

• The vertical axis often represents the quantity that is growing.
• The horizontal axis represents time periods.

At first glance, the curve might look slightly misleading as it could start very flat when the growth rate is low. However, as time progresses, you will notice the curve rising dramatically.
This graphical representation is crucial for identifying exponential growth patterns, especially when predicting future outcomes and making decisions based on growth trends.

Logistic Growth

Logistic growth represents a different process compared to geometric or exponential growth. It describes a situation where growth increases rapidly at first but then slows down as it approaches a maximum limit, often described as the carrying capacity.
A logistic growth model can be described with the equation \( N(t) = \frac{K}{1+\frac{K-N_0}{N_0}e^{-rt}} \), where \( K \) is the carrying capacity, and \( r \) is the growth rate. This model is often used in ecology to represent population growth, where resources such as food or space limit the number of individuals that can be supported.
Real-world examples include populations of animals in a habitat that only has a certain amount of food available or even the spread of diseases, where the number of susceptible hosts limits further spread.

J-shaped Curve

The J-shaped curve is a distinctive feature of exponential growth. It is named because it resembles the letter "J" when graphed.
In the early stages of the growth period, the curve appears flat and slow. But, as time goes on, it rises sharply, showcasing the accelerating pace of exponential growth.
This type of curve is crucial for understanding phenomena like investments, compound interest, and population explosions where there's unchecked and continuous replication or increase. The stark difference between the growth phases of a J-shaped curve highlights why it is potent—it always starts small and slow before rapidly gaining speed.