Exponential Equations - Practice Problems

Solving Earth science problems with exponential growth and decay

Exponential growth

Problem 1. Cyanobacteria, a group of photosynthetic bacteria, have a rapid growth rate that can lead to quickly proliferating populations. In Earth history, cyanobacteria played an essential role in the increase in oxygen in the Earth's atmosphere in the Precambrian, and the organisms are responsible for some algal blooms in lakes.

If a population of cyanobacteria have a growth rate of 0.92 1/(day), how many days will it take for the population to reach 1 billion (1,000,000,000) organisms? Assume the starting population was 5,000 organisms. Remember, the exponential growth equation is

N=N_0 e^(kt)

Problem 2: Determining past population growth and modeling future population growth is important to Earth scientists because society needs to develop plans for resource management, water use, and infrastructure. In 2020, Lagos, Nigeria, one of the fastest growing cities in the world, had a population of 15 million people. If the city continues to grow at rate of 3%, how large will the population be in 2035?

N=N_0 e^(kt)

Exponential decay

Problem 3: A zircon mineral within a granodiorite originally had 1200 mg of 235U and no 207Pb.  It now has 220 mg of 235U.  The decay constant, λ, for the 235U/207Pb system is lambda=9.80xx10^(-10) 1/(yrs). Assuming the only chemical process is the radioactive decay of uranium to lead, what is the age of the rock (which is based on the age of the zircon mineral)?

× Graphing exponential equations

Which type of equation does the graph to the right show?
Exponential growth equations produce curves that increase in slope with increasing x-values

Problem 4. In some cases, you may know the half-life of a radiogenic isotope, but not the decay rate. Thankfully, you can determine the decay rate from the half-life using the age equation. The half-life (t1/2) for the U-Pb system of 238U/206Pb is 4.47 billion years (4,470,000,000 yrs). Use the age equation below to determine the decay constant for 238U. The half life (t1/2) and the decay constant (λ) are related by the equation:

t_(1/2)=ln(2)/lambda

In this problem, we will use the same steps to determine the decay constant from the age equation

N=N_0 e^(-lambdat)

Problem 5. Atmospheric pressure and altitude have an exponential decay relationship describe the barometric equation:

P=P_0*e^(-z/H)

where P is the atmospheric pressure at a given altitude; P0 is the atmospheric pressure at sea level; z is the altitude above sea level; and H is the scale height of the atmosphere, which is the vertical distance pressure decreases by a factor of e (2.718). Using this equation, estimate the atmospheric pressure at an altitude of 4,000 m above sea level. Assume the atmospheric pressure at sea level is 1013 millibars (mb) and the scale height of the atmosphere is 8,000 meters (m).

Problem 6. Ground Penetrating Radar (GPR) is commonly used in geophysical surveys to investigate subsurface features. The attenuation of GPR signals as they penetrate the ground can be modeled using an exponential decay equation:

A=A_0*e^(-kd)

where A is decreased amplitude of the GPR signal; A0 is the initial amplitude of GPR signal; k is the attenuation coefficient; and d is the depth.

Imagine you were tasked with conducting a GPR study in central Minnesota to determine the depth to unmarked utility lines. Your field site is in clay-rich soil that has high conductivity, which means the attenuation coefficient (k) is high. For this clay-rich soil k is 0.4 1/m. If the initial amplitude was 4000 volts per meter (V/m), at what depth will the amplitude be reduced by 50%?

Next steps

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