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Exponential Function Solution Equation - Decay (Continuous) Scenario to Rate

Level 1

This math topic focuses on solving exponential decay problems, specifically calculating the continuous decay rate of populations over time. Each problem provides a scenario where the initial population decreases to a certain amount after a specified period, measured in years, months, or quarters. The task is to rearrange and solve the exponential decay equation to find the rate of decline. Multiple choice solutions are presented with mathematical expressions in LaTeX format, requiring an understanding of logarithmic functions to identify or verify the correct rate formula.

Work on practice problems directly here, or download the printable pdf worksheet to practice offline.

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Exponential Function Solution Equation - Decay (Continuous) Scenario to Rate Worksheet

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Exponential Function Solution Equation - Decay (Continuous) Scenario ...
1
A bacteria population starts at 200. It declines continuously at a certain percent per year. After 6 years it has decreased to a population of 116 bacteria.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{116 over 200 }}{6}
b A LaTex expression showing r = -\ln{\frac{200 over 116 }}{6}
2
A bacteria population starts at 600. It declines continuously at a certain percent per month. After 2 months it has decreased to a population of 501 bacteria.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{600 over 501 }}{2}
b A LaTex expression showing r = -\ln{\frac{501 over 600 }}{2}
c A LaTex expression showing r = -e to the power of \frac{501 over 600 }{2}
3
A bacteria population starts at 800. It declines continuously at a certain percent per month. After 3 months it has decreased to a population of 688 bacteria.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{688 over 800 }}{3}
b A LaTex expression showing r = -\ln{\frac{800 over 688 }}{3}
4
A bacteria population starts at 900. It declines continuously at a certain percent per year. After 5 years it has decreased to a population of 666 bacteria.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{666 over 900 }}{5}
b A LaTex expression showing r = -\ln{\frac{900 over 666 }}{5}
5
A bacteria population starts at 200. It declines continuously at a certain percent per month. After 7 months it has decreased to a population of 162 bacteria.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -e to the power of \frac{162 over 200 }{7}
b A LaTex expression showing r = -\ln{\frac{162 over 200 }}{7}
c A LaTex expression showing r = -\ln{\frac{200 over 162 }}{7}
6
A bird population starts at 800. It declines continuously at a certain percent per quarter. After 4 quarters it has decreased to a population of 604.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{604 over 800 }}{4}
b A LaTex expression showing r = -\ln{\frac{800 over 604 }}{4}
c A LaTex expression showing r = -e to the power of \frac{604 over 800 }{4}
7
A whale population starts at 400. It declines continuously at a certain percent per quarter. After 2 quarters it has decreased to a population of 334 whales.
Rearrange the exponential equation to solve for for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{334 over 400 }}{2}
b A LaTex expression showing r = -\ln{\frac{400 over 334 }}{2}
c A LaTex expression showing r = -e to the power of \frac{334 over 400 }{2}
8
A radioactive material starts at an isotope concentration of 700ppm. It decays continuously at a certain percent per week. After 5 weeks it has decayed to an isotope concentration of 602ppm.
Rearrange the exponential equation to solve for for the rate of decay given this scenario?
a A LaTex expression showing r = -\ln{\frac{602 over 700 }}{5}
b A LaTex expression showing r = -\ln{\frac{700 over 602 }}{5}