what does the a stand for in Pa^t
what does the r stand for in Pe^rt
how are they related?
In the expression Pa^t, the letter "a" represents the initial amount or principal. It typically refers to the starting value or quantity in a mathematical equation or formula.
In the expression Pe^rt, the letter "r" represents the rate of interest or growth rate. It is usually expressed as a decimal or a percentage. This rate determines how fast the value will increase or decrease over time.
The two expressions, Pa^t and Pe^rt, are related in the context of exponential growth or decay. They relate to the future value of an initial amount with the influence of time and growth rate.
The formula Pe^rt is called the exponential growth or decay formula, where:
- P is the initial amount or principal
- e is Euler's number, a mathematical constant approximately equal to 2.71828
- r is the rate of interest or growth rate
- t is the time period
By substituting the appropriate values into the formula and calculating, you can find the future value or amount after a specific time period.
For example, if you have an investment with an initial amount of $1,000 (P = $1,000), an interest rate of 5% (r = 0.05), and you want to find the value after 3 years (t = 3), you can calculate:
Future value = Pe^rt
Future value = 1000e^(0.05*3)
Future value ≈ 1000e^(0.15)
Future value ≈ 1000 * 1.161825
Future value ≈ $1161.83
So, in this case, the future value after 3 years would be approximately $1161.83.