Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
An environmental scientist is modeling the concentration of a pollutant in a lake. The concentration C(t) in parts per million (ppm) after 't' weeks is given by C(t) = 50e^(-0.1t). What is the initial concentration of the pollutant (at t=0)?
- A50 ppm
- B500 ppm
- C0 ppm
- D5 ppm
Show answer & explanationAnswer & explanation
Correct answer: A. 50 ppm
To find the initial concentration, substitute t = 0 into the function. C(0) = 50e^(-0.1 * 0) = 50e^0. Since any non-zero number raised to the power of 0 is 1, e^0 = 1. Therefore, C(0) = 50 * 1 = 50 ppm.
Why the other options are wrong
- B. Incorrect calculation, possibly multiplying 50 by 10.
- C. This would imply no pollutant at the start, which is incorrect.
- D. Incorrect calculation, perhaps misinterpreting the exponent.
Initial Value of Exponential Function
The starting amount or value of a quantity at time t=0 in an exponential growth or decay model.
- In the form y = ab^x, 'a' represents the initial value.
- In the form y = Pe^(rt), 'P' represents the initial value.
- Found by setting the independent variable (often time) to zero and evaluating the function.
Memory trick: Initial means 'time zero', plug it in to find the start!