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Exponential Function Solution Equation - Decay (Continuous) Scenario to Starting Value

Level 1

This math topic centers on practicing rearranging exponential decay equations to solve for starting values of populations or concentrations. The scenarios involve varying decay rates—expressed in percentages per different time units (e.g., quarters, years, weeks, days)—and the resulting values after specified time periods. This topic not only enhances understanding of exponential functions but also hones algebraic manipulation skills in relation to real-world phenomena like population decline and radioactive decay. The problems provide several possible rearrangements of the equation, requiring the identification of the correct form for calculating initial quantities.

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 Starting Value Worksheet

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Exponential Function Solution Equation - Decay (Continuous) Scenario ...
1
A bird population starts at a certain size. It declines continuously at 8% per quarter. After 7 quarters it has decreased to a population of 228.
Rearrange the exponential equation to solve for for the starting population given this scenario?
a A LaTex expression showing P sub 0 = \frac{e to the power of (-0.08 times 7) }{228}
b A LaTex expression showing P sub 0 = 228 over e to the power of (-0.08 times 7)
c A LaTex expression showing P sub 0 = 228 over e to the power of (\frac{-0.08 {7 )}}
2
A whale population starts at a certain size. It declines continuously at 6% per year. After 8 years it has decreased to a population of 433 whales.
Rearrange the exponential equation to solve for for the starting population given this scenario?
a A LaTex expression showing P sub 0 = \frac{e to the power of (-0.06 times 8) }{433}
b A LaTex expression showing P sub 0 = 433 over e to the power of (\frac{-0.06 {8 )}}
c A LaTex expression showing P sub 0 = 433 over e to the power of (-0.06 times 8)
3
A whale population starts at a certain size. It declines continuously at 7% per quarter. After 8 quarters it has decreased to a population of 514 whales.
Rearrange the exponential equation to solve for for the starting population given this scenario?
a A LaTex expression showing P sub 0 = 514 over e to the power of (\frac{-0.07 {8 )}}
b A LaTex expression showing P sub 0 = \frac{e to the power of (-0.07 times 8) }{514}
c A LaTex expression showing P sub 0 = 514 over e to the power of (-0.07 times 8)
4
A bacteria population starts at a certain size. It declines continuously at 8% per week. After 3 weeks it has decreased to a population of 707 bacteria.
Rearrange the exponential equation to solve for for the starting population given this scenario?
a A LaTex expression showing P sub 0 = 707 over e to the power of (\frac{-0.08 {3 )}}
b A LaTex expression showing P sub 0 = \frac{e to the power of (-0.08 times 3) }{707}
c A LaTex expression showing P sub 0 = 707 over e to the power of (-0.08 times 3)
5
A toxin starts at a certain concentration. It declines continuously at 9% per day. After 5 days it has decreased to a concentration of 510mg/L.
Rearrange the exponential equation to solve for for the starting concentration given this scenario?
a A LaTex expression showing C sub 0 = 510 over e to the power of (-0.09 times 5)
b A LaTex expression showing C sub 0 = 510 over e to the power of (\frac{-0.09 {5 )}}
6
A radioactive material starts at a certain isotope concentration. It decays continuously at 3% per week. After 8 weeks it has decayed to an isotope concentration of 157ppm.
Rearrange the exponential equation to solve for for the starting concentration given this scenario?
a A LaTex expression showing R sub 0 = 157 over e to the power of (-0.03 times 8)
b A LaTex expression showing R sub 0 = \frac{e to the power of (-0.03 times 8) }{157}
7
A toxin starts at a certain concentration. It declines continuously at 7% per week. After 4 weeks it has decreased to a concentration of 453mg/L.
Rearrange the exponential equation to solve for for the starting concentration given this scenario?
a A LaTex expression showing C sub 0 = 453 over e to the power of (-0.07 times 4)
b A LaTex expression showing C sub 0 = 453 over e to the power of (\frac{-0.07 {4 )}}
c A LaTex expression showing C sub 0 = \frac{e to the power of (-0.07 times 4) }{453}
8
A toxin starts at a certain concentration. It declines continuously at 4% per day. After 7 days it has decreased to a concentration of 226mg/L.
Rearrange the exponential equation to solve for for the starting concentration given this scenario?
a A LaTex expression showing C sub 0 = 226 over e to the power of (\frac{-0.04 {7 )}}
b A LaTex expression showing C sub 0 = 226 over e to the power of (-0.04 times 7)
c A LaTex expression showing C sub 0 = \frac{e to the power of (-0.04 times 7) }{226}