An exponential model multiplies its starting amount by the same factor each hour. If you’re given a later count, substitute its time and count into the model, then use a Desmos regression to find the hourly factor. A factor applied over several hours is a power of the hourly factor, so don’t mistake the whole-period growth for growth in one hour.
Hints
- Hint 1
The exponent counts how many times the hourly factor has been applied. After three hours, that’s three multiplications by . What equation do you get when you substitute the recorded time and count?
- Hint 2
A growth factor is the multiplier for one hour, not for all three hours together. Use in Desmos to find from your equation, then round its value to match the choices.
Step-by-step
Find the hourly factor with Desmos
Step 1Use the three-hour count
At hours, the factor has been applied three times, giving . Substitute the recorded count into the given model: . The ratio of the two counts covers three hours, so it isn’t itself.
- Step 2
Solve for the hourly factor
Type in Desmos. The tells Desmos to find the unknown growth factor; under PARAMETERS, it shows . This is the hourly factor because the model raises to the number of hours.
- Step 3
Round to match the choices
The choices give factors to the nearest hundredth. Since has a in the third decimal place, it rounds to , not .
- Step 4
Keep the starting count
Keep as the initial value, the count at , because . is the count after three hours, not at the start. So the model is . Choice D.