Quick Answer
Exponential decay is a process where a quantity decreases by a consistent percentage over equal time intervals. On the Digital SAT, this concept typically appears in the Math section within the Passport to Advanced Math domain, appearing in approximately 2–4 questions per test to model real-world depreciation or population decline.
Exponential decay is a mathematical relationship where a value reduces at a rate proportional to its current size, expressed by the formula y = a(1 - r)^t or y = ab^x where 0 < b < 1.
Question: A radioactive substance with an initial mass of 500 grams decays at a rate of 15% per hour. Which function M(t) models the mass remaining after t hours? Solution: Use the formula M(t) = a(1 - r)^t. Here, a = 500 and r = 0.15. Thus, M(t) = 500(1 - 0.15)^t, which simplifies to M(t) = 500(0.85)^t.
Using the decay rate instead of the decay factor: Students often mistakenly use the percentage (e.g., 0.15) as the base of the exponent rather than subtracting it from 1 (e.g., 0.85).
Confusing linear and exponential decay: Students may attempt to subtract a constant amount each period instead of multiplying by a constant ratio, resulting in an incorrect linear model.
Incorrect time interval scaling: Students sometimes fail to adjust the exponent when the decay rate is given for a time period different from the variable t, such as a rate per decade when t is in years.
Students targeting 750+ should know that the SAT may present exponential decay using fractional exponents, such as t/k, where k represents the time it takes for a specific amount of decay to occur. Recognizing that the base must be less than 1 regardless of the exponent's complexity is key to quickly eliminating incorrect answer choices.
Exponential Function
An exponential function is a mathematical relationship where a constant ratio determines the change in the dependent variable. On the Digital SAT, these functions frequently appear in the Math section, specifically within the Advanced Math domain, often requiring students to interpret growth or decay constants in real-world modeling word problems.
Exponent
An exponent indicates the number of times a base is multiplied by itself. On the Digital SAT, exponent rules are a fundamental component of the Math section, appearing in approximately 10-15% of questions. Students typically encounter these within the Passport to Advanced Math and Heart of Algebra categories.
Exponential Growth
Exponential growth is a process where a quantity increases by a fixed percentage over equal time intervals. On the Digital SAT, this concept appears frequently in the Math Section (Modules 1 and 2), typically within 'Passport to Advanced Math' or 'Problem Solving and Data Analysis' question types requiring equation interpretation or modeling.
Exponential decay on the SAT refers to mathematical models where a quantity decreases by a fixed percentage over time. It is a core concept in the Math section, often tested through word problems involving depreciation or biological half-lives. Students must be able to identify, construct, and interpret functions where the base of the exponent is a value between 0 and 1.
To calculate exponential decay, use the formula y = a(1 - r)^t. First, identify the initial amount (a) and the rate of decrease (r) expressed as a decimal. Subtract the rate from 1 to find the decay factor, then raise this factor to the power of the time elapsed (t). Multiplying this result by the initial amount gives the final value.
The primary difference lies in the base of the exponent, known as the growth or decay factor. In exponential growth, the base is greater than 1 (1 + r), causing the total value to increase over time. In exponential decay, the base is between 0 and 1 (1 - r), causing the total value to decrease toward a horizontal asymptote, usually zero.
Typically, the Digital SAT includes approximately 2 to 4 questions specifically focused on exponential models, which include both growth and decay. While the exact number varies by test form, these questions are a staple of the 'Passport to Advanced Math' category and are essential for students aiming for a high math score.
