In an exponential model, the base is the multiplier for one full growth step, and the exponent counts those steps. If a question gives the percent growth and asks how often it happens, use a Desmos regression to find when the output first reaches the starting size times the growth factor. Then convert that time to the requested units. A number in the exponent is not necessarily the length of the interval.
Hints
- Hint 1
A growth factor includes the original colony. A increase means multiply the starting size by , not . What size is one increase past the starting size?
- Hint 2
The exponent counts how many times the multiplier has been applied. One full increase happens when the exponent reaches . What value of makes that happen?
- Hint 3
The model measures in years, while the question asks for months. Once you find the time for one increase, use months per year to convert it.
Step-by-step
Find one growth interval with Desmos
Step 1Find the starting size
Type , then . Desmos displays , in thousands. This is the starting size, the size at , because the exponent is then. The measures bacteria, not months.
- Step 2
Find the size after one increase
A growth factor of keeps the original and adds . So one full increase makes the target size , not . Type ; Desmos displays thousand.
- Step 3
Find when the colony reaches that size
Type . The subscript makes a value Desmos can find, and tells it to match the two sides. Under PARAMETERS, Desmos shows . That's years for one increase: at that time, the exponent reaches , so the factor has been applied once. The itself isn't a time interval.
- Step 4
Convert the interval to months
The model counts years, so use months per year. Type ; Desmos displays . The colony is predicted to increase by every months. Find when the exponent counts one full growth step, then convert the time units. Choice D.