Question

In Exercises 27-30, use the properties of exponents to rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then find the percent rate of change. $$ y=e^{-0.25 t} $$

Short answer

The function in the form \(y=a(1+r)^t\) is \(y=1(1-0.25)^t\). The percent rate of change (or interest rate) is -25%.

Step-by-step solution

Rewrite the Function

To begin with, notice that the given function \(y=e^{-0.25 t}\) can be rewritten in the form \(y=a(1+r)^t\). In this case, \(a=1\), and \(r=-0.25\). This is because the base of the exponent is Euler's number \(e\), and we can write this as a compound continuously function.

Find the Equivalent Form

The given exponential function is already in the form of \(y=a(1+r)^t\). Hence, it can be rewritten as \(y=1(1-0.25)^t\). This form shows us that the quantity '(1-0.25)' plays the role as an equivalent base here.

Calculate the Percentage Rate of Change

The rate of change, \(r\), has been found above to be -0.25. To express this as a percentage, multiply by 100. So the percent rate of change is \(-0.25 \times 100 = -25\%\). It means that the function is decreasing by 25%.

Interpretation of the Result

The negative sign indicates a rate of decrease. That is, the function decreases by 25% every time \(t\) increases by 1.

Key concepts

Properties of Exponents

Understanding the properties of exponents is essential for solving various algebraic problems, particularly when working with exponential functions. Exponents, also known as powers, are shorthand for repeated multiplication. For example, when we write 3 to the power of 2, denoted as 3^2, we mean 3 multiplied by itself, which equals 9.

The main properties of exponents include the product rule, power rule, quotient rule, and negative exponent rule. Here's a brief explanation of each:

• Product rule: When multiplying two powers that have the same base, you can add the exponents. For example, x^a \( \cdot \) x^b = x^{(a+b)}.
• Power rule: When raising a power to another power, you can multiply the exponents. For instance, (x^a)^b = x^{(a \cdot b)}.
• Quotient rule: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. For example, x^a / x^b = x^{(a-b)}.
• Negative exponent rule: A negative exponent means you take the reciprocal of the base and make the exponent positive. For example, x^-a = 1/x^a.

These properties allow you to manipulate and simplify expressions with exponents, making it easier to work with equations and functions.

Percent Rate of Change

The percent rate of change is a concept that expresses how much a quantity changes as a percentage over a specific period. It's crucial in contexts where we are dealing with growth or decay, such as population studies, finance, or radioactive decay.

To find the percent rate of change, you typically follow these steps:

• Identify the initial quantity and the new quantity.
• Calculate the change by subtracting the initial quantity from the new quantity.
• Divide the change by the initial quantity to get the rate of change.
• Multiply the rate of change by 100 to convert it to a percentage.

For example, if a bank account balance increases from \(1,000 to \)1,100 over a year, the percent rate of change would be ((\(1,100 - \)1,000) / $1,000) \( \times \) 100, which equals 10%. This means the bank account saw a 10% increase over the year.

Euler's Number

Euler's number, often denoted as \(e\), is an irrational constant that is approximately equal to 2.71828. It is one of the most important numbers in mathematics, particularly within the context of exponential growth and decay, compound interest, and certain probability distributions.

Euler's number is the base of natural logarithms and is unique because the function \(e^x\) is equal to its own derivative. In simpler terms, this means if you were to graph the function \(y = e^x\), the slope of the curve at any point is equal to the value of \(y\) at that point. This property makes it incredibly useful in calculations involving continuous growth or decay. When you encounter an exponential function with \(e\) as the base, such as \(y=e^{-0.25 t}\), you're often dealing with a situation that involves continuous processes, such as radioactive decay, population growth, or compound interest calculated continuously over time.