Problem 46
Question
The area of Brazil's rain forest, \(R=f(t),\) in million acres, is a function of the number of years, \(t,\) since 2000 (a) Interpret \(f(9)=740\) and \(f^{\prime}(9)=-2.7\) in terms of Brazil's rain forests. 16 (b) Find and interpret the relative rate of change of \(f(t)\) when \(t=9\)
Step-by-Step Solution
Verified Answer
In 2009, Brazil's rain forest was 740 million acres and decreasing by 0.365% annually.
1Step 1: Understand f(9)=740
The expression \( f(9) = 740 \) represents the area of Brazil's rain forest 9 years after 2000. Therefore, the area of the rain forest in 2009 is 740 million acres.
2Step 2: Interpret f'(9)=-2.7
The derivative \( f'(t) \) gives us the rate of change of the area with respect to time. Therefore, \( f'(9) = -2.7 \) means that in the year 2009, the area of Brazil's rain forest was decreasing at a rate of 2.7 million acres per year.
3Step 3: Find Relative Rate of Change
The relative rate of change at \( t = 9 \) is given by \( \frac{f'(9)}{f(9)} \). Therefore, we have: \[ \frac{-2.7}{740} \approx -0.00365 \].
4Step 4: Interpret Relative Rate of Change
The relative rate of change \( -0.00365 \) implies that at \( t = 9 \) (or in 2009), the area of Brazil's rain forest was decreasing by approximately 0.365% per year.
Key Concepts
DerivativeRate of ChangeRelative Rate of Change
Derivative
The **derivative**, often symbolized by \( f'(t) \), is a fundamental concept in calculus. It reflects how a function changes as its input changes. Think of it as a "mathematical measurement" of how fast or slow something is changing at a specific moment.
In our exercise, the derivative \( f'(9) = -2.7 \) provides the rate of change of the rainforest's area, in million acres per year, when \( t = 9 \). A negative value indicates a decrease. Hence, in 2009, Brazil's rainforest was shrinking by approximately 2.7 million acres each year.
Derivatives are crucial because they offer insights into the behavior of functions, helping us predict trends and make informed decisions. Understanding them helps analyze real-world scenarios, like environmental changes, growth rates, and more.
In our exercise, the derivative \( f'(9) = -2.7 \) provides the rate of change of the rainforest's area, in million acres per year, when \( t = 9 \). A negative value indicates a decrease. Hence, in 2009, Brazil's rainforest was shrinking by approximately 2.7 million acres each year.
Derivatives are crucial because they offer insights into the behavior of functions, helping us predict trends and make informed decisions. Understanding them helps analyze real-world scenarios, like environmental changes, growth rates, and more.
Rate of Change
Rate of Change represents how a quantity changes over time. Much like speed measures how fast a car travels, rate of change tells us how quickly the value of a function varies.
In our example, the area of Brazil's rainforest changes over the years. The derivative \( f'(t) \) gives us that rate for any given time \( t \). Specifically, \( f'(9) = -2.7 \) tells us that in 2009, the rainforest area was decreasing by 2.7 million acres annually.
Understanding the rate of change is essential in many fields:
In our example, the area of Brazil's rainforest changes over the years. The derivative \( f'(t) \) gives us that rate for any given time \( t \). Specifically, \( f'(9) = -2.7 \) tells us that in 2009, the rainforest area was decreasing by 2.7 million acres annually.
Understanding the rate of change is essential in many fields:
- Helps identify trends
- Supports effective planning and resource allocation
- Provides a basis for data-driven decisions
Relative Rate of Change
The **relative rate of change** offers a perspective on how significant the change is compared to the overall size or quantity. While the rate of change gives an absolute value, the relative rate places this in context as a proportion.
Mathematically, it is calculated as the ratio of the derivative to the function's value: \( \frac{f'(9)}{f(9)} \). In the exercise, this is \( \frac{-2.7}{740} \approx -0.00365 \), which translates to a 0.365% decrease in the rainforest area per year at \( t = 9 \).
This measure puts the change into perspective:
Mathematically, it is calculated as the ratio of the derivative to the function's value: \( \frac{f'(9)}{f(9)} \). In the exercise, this is \( \frac{-2.7}{740} \approx -0.00365 \), which translates to a 0.365% decrease in the rainforest area per year at \( t = 9 \).
This measure puts the change into perspective:
- It shows how substantial the annual change is relative to the total area
- Makes it easier to compare changes across different scales
- Useful for economic, scientific, and financial analyses
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