ACT Preparation / Unit 12 ACT 1104 worked example, 60 to 90 seconds
Writing a Model From a Percent Change
The idea
An exponential model is written from a description in three steps. First, find the initial value a, the amount at time zero. Second, convert the percent change to a growth factor: 1 plus the rate for growth, 1 minus the rate for decay. Third, write f(t) = a·bᵗ, checking that t counts the same unit the rate is given per, and evaluate it if asked.
In practice
Worked example: a town of 12,000 grows 3% a year. a = 12,000 and b = 1.03, so P(t) = 12,000(1.03)ᵗ. After 2 years: 12,000 × 1.0609 = 12,730.8, about 12,731. Second: a car worth $25,000 loses 15% of its value each year: V(t) = 25,000(0.85)ᵗ, so after 2 years it is worth 25,000 × 0.7225 = $18,062.50.
Where people slip
Each year's percent applies to the new amount, so two years of 3% growth is a factor of 1.0609, not 1.06. This is unit seven's rule for successive changes, written as a function.
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