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Chapter 4: Problem 36

It has been estimated that the world's tropical rain forests are disappearing at the rate of \(1.8 \%\) per year. If this rate continues, how soon will the rain forests be reduced to \(50 \%\) of their present size? (Rain forests not only generate much of the oxygen that we breathe but also contain plants with unique medical properties, such as the rosy periwinkle which has revolutionized the treatment of leukemia.)

Short Answer

Expert verified
In approximately 38 years, the rain forests will be reduced to 50% of their current size.

Step by step solution

01

Understand the Problem

We need to find out in how many years the size of the tropical rain forests will be reduced to 50% of their current size if they are decreasing by 1.8% annually. This is a problem of exponential decay.
02

Set Up the Exponential Decay Formula

The formula for exponential decay is given by \( A = P (1 - r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time in years. We need to solve for \( t \).
03

Assign Known Values

Here, \( A = 0.5P \), \( r = 0.018 \), and we need to find \( t \). The equation becomes \( 0.5P = P (1 - 0.018)^t \).
04

Simplify the Equation

Divide both sides of the equation by \( P \) to simplify: \( 0.5 = (1 - 0.018)^t \). This gives us \( 0.5 = 0.982^t \).
05

Solve for Time \( t \)

To solve for \( t \), take the natural logarithm of both sides: \( \ln(0.5) = \ln(0.982^t) \). This simplifies to \( \ln(0.5) = t \cdot \ln(0.982) \).
06

Isolate \( t \)

Divide both sides by \( \ln(0.982) \) to solve for \( t \): \( t = \frac{\ln(0.5)}{\ln(0.982)} \).
07

Calculate \( t \)

Calculate the values: \( \ln(0.5) \approx -0.6931 \) and \( \ln(0.982) \approx -0.0182 \). Now compute \( t = \frac{-0.6931}{-0.0182} \approx 38.081 \).
08

Conclusion

Since \( t \approx 38.081 \), the tropical rain forests will be reduced to 50% of their current size in approximately 38 years.

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions describe processes that change rapidly, such as population growth, radioactive decay, and investment growth. They are characterized by their self-reinforcing nature: the bigger the function, the faster it grows or decays.
In the context of exponential decay, the formula is given by \( A = P(1 - r)^t \). Here, \( A \) represents the future quantity, \( P \) is the starting amount, \( r \) is the decay rate, and \( t \) is the time period.
This mathematical model helps us understand real-world phenomena like the decline of tropical rainforests. By applying exponential functions, we can predict how soon an ecosystem, currently shrinking by a specific percentage each year, will reach a particular size.
Tropical Rainforests
Tropical rainforests are lush, biodiverse ecosystems found in equatorial regions. These forests are crucial to the earth's health, known for their role in oxygen production and carbon dioxide absorption.
Features of tropical rainforests include:
  • High rainfall: They receive over 200 cm of rain annually, sustaining a wide range of flora and fauna.
  • Biodiversity: Home to half of the world's plant and animal species, they offer untapped medicinal and ecological resources.
  • Climate regulation: By absorbing carbon dioxide, they help control global temperatures.
However, they are under threat from deforestation, driven by agricultural expansion, logging, and infrastructure development. As these forests dwindle, we not only lose biodiversity but also compromise our planet's ability to regulate climate conditions.
Environmental Mathematics
Environmental mathematics combines mathematical techniques with environmental concerns to solve complex ecological issues. It is instrumental in addressing matters like climate change, resource management, and biodiversity conservation.
In the study of ecosystems such as rainforests, mathematical models like exponential decay equations provide insight into the long-term impacts of human activities.
Using the exponential decay formula, environmental scientists can assess:
  • The rate at which rainforests are being cleared.
  • The future size of remaining forests.
  • Potential impacts on local and global climates.
By understanding these dynamics, policymakers can devise strategies to mitigate deforestation, ensuring sustainable management of natural resources.

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Most popular questions from this chapter

97-98. ATHLETICS: World's Record 100-Meter Run In 1987 Carl Lewis set a new world's record of 9.93 seconds for the 100 -meter run. The distance that he ran in the first \(x\) seconds was $$ 11.274\left[x-1.06\left(1-e^{-x / 1.06}\right)\right] \text { meters } $$ for \(0 \leq x \leq 9.93 .\) Enter this function as \(y_{1},\) and define \(y_{2}\) as its derivative (using NDERIV), so that \(y_{2}\) gives his velocity after \(x\) seconds. Graph them on the window [0,9.93] by [0,100] Trace along the velocity curve to verify that Lewis's maximum speed was about 11.27 meters per second. Find how quickly he reached a speed of 10 meters per second, which is \(95 \%\) of his maximum speed.

Between 2009 and 2010 , the number of satellite radio subscribers in the United States increased by \(9 \%\). At that rate, when will the number increase by \(50 \% ?\)

The time required for the amount of a drug in one's body to decrease by half is called the "half-life" of the drug. a. For a drug with absorption constant \(k\), derive the following formula for its half-life: $$ \left(\begin{array}{c} \text { Half- } \\ \text { life } \end{array}\right)=\frac{\ln 2}{k} $$

Recall that the concentration of a drug in the bloodstream after \(t\) hours is \(c e^{-k t},\) where \(k\) is called the "absorbtion constant." If one drug has a larger absorbtion constant than another, will it require more or less time between doses? (Assume that both drugs have the same value of \(c .)\)

Find a formula for the time required for an investment to grow to \(k\) times its original size if it grows at interest rate \(r\) compounded annually.
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