Problem 15 - Elementary and Intermediate Algebra

Question

Match each situation to the exponential graph that best models it. a. The number of cell phone subscribers in the world over the past 5 years b. The level of caffeine in the bloodstream after drinking a cup of coffee c. The amount of money in a bank account earning interest compounded quarterly d. The number of rabbits in a population with a high birth rate e. The amount of water in a shirt that was just washed and hung on a clothesline to dry (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

A: Rapid growth; B: Decline; C: Steady growth; D: Rapid growth; E: Decline.

Step 1: Understand the Contexts

Before matching, recognize the nature of each scenario: - A: A rapidly growing system (cell phone subscribers). - B: A system that decreases over time (caffeine levels). - C: A financial system with periodic growth (bank interest). - D: Biologically rapid growth (rabbit population). - E: Gradual reduction over time (water content).

Step 2: Identify Graph Characteristics

Matching each context to an exponential graph involves understanding two main graph forms: - Exponential growth, where quantity increases rapidly (graphs for A, C, D). - Exponential decay, where quantity decreases to zero (graphs for B, E).

Step 3: Assign Graphs Based on Growth

- Rapid increase over time: Matches graphs with steep upward trends. - Slowly or periodically growing (C): Matches a less steep upward graph, possibly periodic based. - Rapid increase in population (A&D): Likely to match steeper growth graphs.

Step 4: Assign Graphs Based on Decay

- Caffeine decrease (B): Matches graphs where the curve descends rapidly. - Water content decrease (E): Matches graphs with a gradual downward slope.

Step 5: Final Matching

A: Steep exponential growth graph, likely trend replicating. B: Sharply declining graph, caffeine diminishes quickly. C: Periodic/steady growing curve, matching bank interest growth. D: Another rapid growth curve like A, representing a high birth rate. E: Gradual downward slope, suitable for drying wet clothing.

Key Concepts

Exponential Growth

Exponential Growth is a fascinating phenomenon depicting how things expand rapidly over time. Imagine you are watching a tiny snowball rolling down a hill and growing larger with each turn. This is what exponential growth looks like. When we apply this concept to real-world scenarios, we often see it in populations or technological advancements. For instance, the number of cell phone subscribers in the world is a perfect example. As time progresses, more people subscribe, leading to the numbers shooting up rapidly.

The mathematical representation of exponential growth is typically given by the formula \( P(t) = P_0 \cdot e^{rt} \), where \( P(t) \) is the amount of quantity at time \( t \), \( P_0 \) is the initial quantity, \( r \) is the growth rate, and \( e \) is Euler’s number (approximately 2.718). This formula helps us visualize and predict how quickly a quantity can grow if left unchecked.

• It's essential in understanding things like bacteria growth, human populations, and technology usage.
• Everything doubling at regular intervals is a hallmark of exponential growth.

In our exercise, the scenarios involving rapid increases, such as cell phone subscribers and rabbit populations due to high birth rates, best illustrate exponential growth on graphs.

Exponential Decay

Exponential Decay is the opposite of exponential growth. It's like watching a pile of ice slowly melt until it's all gone. Instead of growing, the quantity decreases over time, typically at a rapid pace initially before slowing down. A common example of exponential decay is the level of caffeine in the bloodstream after drinking coffee. Once ingested, caffeine levels peak and then fall off quickly as the body metabolizes it.

The formula used to describe exponential decay has a similar form to growth: \( N(t) = N_0 \cdot e^{-kt} \), where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay rate, and \( e \) is Euler’s number.

• Exponential decay effectively models the natural reduction of substances.
• It's used in radioactive decay, depleting resources, and absorption of substances in the human body.

In our exercise, the gradual decline of caffeine levels in the body and the decrease in water content as a shirt dries can be observed through graphs showcasing exponential decay.

Graph Interpretation

Interpreting graphs correctly is key to understanding exponential functions in real life. In exponential growth graphs, you'll notice a steep upward curve, often starting slow and becoming more rapid. Imagine climbing a hill that becomes steeper the higher you go; that’s how these graphs appear.

On the other hand, exponential decay graphs begin with a fast drop, followed by a leveling off. Like a car quickly decelerating at a stop sign, the rate of decline decreases after the initial plunge.

• A graph depicting exponential growth might represent rapidly increasing data like populations or investments.
• Conversely, exponential decay is portrayed as a downward slope indicating substance reduction over time.

Proper interpretation allows us to match scenarios with their respective graphs accurately, such as associating high birth rates with sharp increase graphs or caffeine decay with sharp decline graphs.

Mathematical Modeling

Mathematical Modeling using exponential functions allows us to simulate real-world scenarios. By creating a model, we transform complex systems into understandable equations and graphs.

In exponential growth models, we estimate things like financial investments where interest is compounded. The bank account earning interest semi-annually fits into this category. These models help in planning and predicting future outcomes.

For exponential decay, models predict how quickly substances or populations decline, such as medications in a patient’s body or drying processes like the shirt on a clothesline.

• These models are crucial in resource planning, predicting market trends, and scientific simulations.
• They enable us to use mathematical reasoning for solving practical problems.

In our exercise, identifying scenarios such as rabbits' population growth or caffeine reduction helps us apply suitable exponential models for accurate graph matching.