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Exponential Function Solving - Decay (Continuous) - Scenario to Time

Level 1

This math topic focuses on solving for time in continuous decay scenarios using exponential functions. Skills practiced include: 1. Understanding and applying the exponential decay model in the form of \( P(t) = P_0 \times e^{rt} \), where \( P(t) \) represents the quantity at time \( t \), \( P_0 \) is the initial quantity, \( r \) is the rate of decay, and \( e \) is the base of the natural logarithm. 2. Calculating the time required for a population or quantity, like whale populations, radioactive isotopes, or bacteria, to decrease from an initial value to a lower value at a given continuous decay rate. 3. Manipulating algebraic and logarithmic expressions to isolate and solve for the variable of interest, which is time in these problems.

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 Time Worksheet

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Exponential Function Solving - Decay (Continuous) - Scenario to Time
1
A whale population starts at 200. It declines continuously at 9% per year. After a certain number of years it has decreased to a population of 127 whales.
Solve for the time given this scenario?
a A LaTex expression showing t = -\ln{\frac{P over P sub 0 }}{r}
b A LaTex expression showing 8 + t = -\frac{\ln{P times P sub 0 }}{r}
c A LaTex expression showing 7 + t = -\frac{\ln{P times P sub 0 }}{r}
d A LaTex expression showing 3 + t = -r over \ln{\frac{P {P sub 0 }}}
2
A radioactive material starts at an isotope concentration of 300ppm. It decays continuously at 9% per day. After a certain number of days it has decayed to an isotope concentration of 209ppm.
Solve for the time given this scenario?
a A LaTex expression showing 3 + t = -\frac{\ln{R times R sub 0 }}{r}
b A LaTex expression showing 3 + t = -r over \ln{\frac{R {R sub 0 }}}
c A LaTex expression showing 6 + t = -r over \ln{\frac{R {R sub 0 }}}
d A LaTex expression showing t = -\ln{\frac{R over R sub 0 }}{r}
3
A bacteria population starts at 700. It declines continuously at 9% per day. After a certain number of days it has decreased to a population of 340 bacteria.
Solve for the time given this scenario?
a A LaTex expression showing 1 + t = -\frac{\ln{P times P sub 0 }}{r}
b A LaTex expression showing t = -\ln{\frac{P over P sub 0 }}{r}
c A LaTex expression showing 3 + t = -r over \ln{\frac{P {P sub 0 }}}
d A LaTex expression showing 2 + t = -\frac{\ln{P times P sub 0 }}{r}
4
A whale population starts at 200. It declines continuously at 9% per year. After a certain number of years it has decreased to a population of 116 whales.
Solve for the time given this scenario?
a A LaTex expression showing 6 + t = -\frac{\ln{P times P sub 0 }}{r}
b A LaTex expression showing 3 + t = -\frac{\ln{P times P sub 0 }}{r}
c A LaTex expression showing 1 + t = -\frac{\ln{P times P sub 0 }}{r}
d A LaTex expression showing t = -\ln{\frac{P over P sub 0 }}{r}
5
A bacteria population starts at 800. It declines continuously at 4% per month. After a certain number of months it has decreased to a population of 738 bacteria.
Solve for the time given this scenario?
a A LaTex expression showing t = -\ln{\frac{P over P sub 0 }}{r}
b A LaTex expression showing 2 + t = -\frac{\ln{P times P sub 0 }}{r}
c A LaTex expression showing 7 + t = -\frac{\ln{P times P sub 0 }}{r}
d A LaTex expression showing 2 + t = -r over \ln{\frac{P {P sub 0 }}}
6
A radioactive material starts at an isotope concentration of 600ppm. It decays continuously at 3% per week. After a certain number of weeks it has decayed to an isotope concentration of 532ppm.
Solve for the time given this scenario?
a A LaTex expression showing 3 + t = -\frac{\ln{R times R sub 0 }}{r}
b A LaTex expression showing 7 + t = -\frac{\ln{R times R sub 0 }}{r}
c A LaTex expression showing 6 + t = -\frac{\ln{R times R sub 0 }}{r}
d A LaTex expression showing t = -\ln{\frac{R over R sub 0 }}{r}
7
A radioactive material starts at an isotope concentration of 300ppm. It decays continuously at 6% per hour. After a certain number of hours it has decayed to an isotope concentration of 174ppm.
Solve for the time given this scenario?
a A LaTex expression showing 9 + t = -\frac{\ln{R times R sub 0 }}{r}
b A LaTex expression showing 6 + t = -\frac{\ln{R times R sub 0 }}{r}
c A LaTex expression showing 4 + t = -\frac{\ln{R times R sub 0 }}{r}
d A LaTex expression showing t = -\ln{\frac{R over R sub 0 }}{r}
8
A bacteria population starts at 900. It declines continuously at 2% per year. After a certain number of years it has decreased to a population of 847 bacteria.
Solve for the time given this scenario?
a A LaTex expression showing 3 + t = -\frac{\ln{P times P sub 0 }}{r}
b A LaTex expression showing t = -\ln{\frac{P over P sub 0 }}{r}
c A LaTex expression showing 7 + t = -r over \ln{\frac{P {P sub 0 }}}
d A LaTex expression showing 5 + t = -\frac{\ln{P times P sub 0 }}{r}