An exponential model multiplies a starting amount by the same factor each time interval; the exponent counts those intervals. If you know a later amount but need the starting amount, substitute the given time and population, then use a one-unknown Desmos regression. Multiplying the later amount by the growth factor moves forward, not back.
Hints
- Hint 1
In an exponential model, the exponent counts how many times the growth factor is applied. The given population is measured after growth. What equation connects that population to the starting amount?
- Hint 2
The initial value is the amount before any growth. To work backward, undo the multiplication by the growth factor for all the hours that have passed, not just one hour.
Step-by-step
Substitute the known population and solve
Step 1Turn the measurement into an equation
The colony has bacteria when , so . Put those values into the given model:
The exponent means the growth factor has been applied three times.
- Step 2
Find the starting population
To work backward through three rounds of growth, divide by rather than multiply by it. Type in Desmos. The regression symbol asks Desmos to find the missing constant; under PARAMETERS, it shows . So the colony initially contained bacteria. Grid in 128.