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Exponential Function Solving - Decay (Continuous) Scenario to Rate

Level 1

This math topic focuses on solving for the decay rate in continuous exponential decay models. The questions involve scenarios where the initial quantity decreases continuously over time, and students must find the percentage decline per year, week, quarter, or other duration. The problems cover various contexts such as wildlife populations, bacteria levels, radioactive isotopes, and toxin concentrations. Each scenario requires applying logarithmic and exponential manipulations to derive the decay rate based on given initial and final values over a specified time period.

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

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Exponential Function Solving - Decay (Continuous) Scenario to Rate Worksheet

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Exponential Function Solving - Decay (Continuous) Scenario to Rate
1
A whale population starts at 200. It declines continuously at a certain percent per year. After 7 years it has decreased to a population of 114 whales.
Solve for the rate given this scenario?
a A LaTex expression showing 9 + r = -e to the power of \frac{P over P sub 0 }{t}
b A LaTex expression showing 3 + r = -e to the power of \frac{P over P sub 0 }{t}
c A LaTex expression showing r = -\ln{\frac{P over P sub 0 }}{t}
2
A bacteria population starts at 900. It declines continuously at a certain percent per week. After 7 weeks it has decreased to a population of 514 bacteria.
Solve for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{P over P sub 0 }}{t}
b A LaTex expression showing 1 + r = -\ln{\frac{P sub 0 over P }}{t}
c A LaTex expression showing 0 + r = -e to the power of \frac{P over P sub 0 }{t}
d A LaTex expression showing 8 + r = -\ln{\frac{P sub 0 over P }}{t}
3
A radioactive material starts at an isotope concentration of 300ppm. It decays continuously at a certain percent per hour. After 5 hours it has decayed to an isotope concentration of 211ppm.
Solve for the rate of decay given this scenario?
a A LaTex expression showing 5 + r = -\ln{\frac{R sub 0 over R }}{t}
b A LaTex expression showing 1 + r = -\ln{\frac{R sub 0 over R }}{t}
c A LaTex expression showing r = -\ln{\frac{R over R sub 0 }}{t}
d A LaTex expression showing 8 + r = -\ln{\frac{R sub 0 over R }}{t}
4
A bird population starts at 800. It declines continuously at a certain percent per quarter. After 7 quarters it has decreased to a population of 426.
Solve for the rate given this scenario?
a A LaTex expression showing 5 + r = -e to the power of \frac{P over P sub 0 }{t}
b A LaTex expression showing 6 + r = -e to the power of \frac{P over P sub 0 }{t}
c A LaTex expression showing 7 + r = -\ln{\frac{P sub 0 over P }}{t}
d A LaTex expression showing r = -\ln{\frac{P over P sub 0 }}{t}
5
A radioactive material starts at an isotope concentration of 300ppm. It decays continuously at a certain percent per day. After 7 days it has decayed to an isotope concentration of 197ppm.
Solve for the rate of decay given this scenario?
a A LaTex expression showing 4 + r = -e to the power of \frac{R over R sub 0 }{t}
b A LaTex expression showing 3 + r = -\ln{\frac{R sub 0 over R }}{t}
c A LaTex expression showing 8 + r = -e to the power of \frac{R over R sub 0 }{t}
d A LaTex expression showing r = -\ln{\frac{R over R sub 0 }}{t}
6
A toxin starts at a concentration of 400mg/L. It declines continuously at a certain percent per day. After 7 days it has decreased to a concentration of 228mg/L.
Solve for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{C over C sub 0 }}{t}
b A LaTex expression showing 9 + r = -e to the power of \frac{C over C sub 0 }{t}
c A LaTex expression showing 2 + r = -\ln{\frac{C sub 0 over C }}{t}
d A LaTex expression showing 4 + r = -\ln{\frac{C sub 0 over C }}{t}
7
A toxin starts at a concentration of 800mg/L. It declines continuously at a certain percent per month. After 3 months it has decreased to a concentration of 709mg/L.
Solve for the rate given this scenario?
a A LaTex expression showing r = -\ln{\frac{C over C sub 0 }}{t}
b A LaTex expression showing 9 + r = -\ln{\frac{C sub 0 over C }}{t}
c A LaTex expression showing 7 + r = -\ln{\frac{C sub 0 over C }}{t}
d A LaTex expression showing 2 + r = -e to the power of \frac{C over C sub 0 }{t}
8
A whale population starts at 200. It declines continuously at a certain percent per year. After 3 years it has decreased to a population of 167 whales.
Solve for the rate given this scenario?
a A LaTex expression showing 4 + r = -\ln{\frac{P sub 0 over P }}{t}
b A LaTex expression showing r = -\ln{\frac{P over P sub 0 }}{t}
c A LaTex expression showing 9 + r = -e to the power of \frac{P over P sub 0 }{t}