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. $$ y=0.5 e^{0.8 t} $$

Short answer

The function in the form \(y = a(1 + r)^t\) is \(y = 0.5(1 + r)^t\) where \(r = 2.71828^{0.8} - 1\). The percent rate of change would be \(r * 100\%\).

Step-by-step solution

Reformat the given function

First, rewrite the given function in terms of \(e\), which approximately equals to 2.71828. So, the function becomes \(y=0.5*(2.71828)^{0.8 t}\)

Rewrite the function in the form y=a(1+r)^t

The exponential growth function can be rewritten in the form \( y = a(1+r)^t \), where - 'a' is the current value, 'r' is the growth rate, and 't' is time. By comparing the modified function to this form, we can say \(a=0.5\) (the current quantity or initial value), and \((1+r)=2.71828^{0.8}\) since the base 'e' to the power of 0.8 is equal to \((1+r)\).

Solve for 'r'

Now solve for 'r' from \((1+r)=2.71828^{0.8}\) which gives, \(r=2.71828^{0.8} - 1\)

Find the percent rate of change

To find the percent rate of change, multiply 'r' by 100. So, the percent rate of change = \(r * 100\%\)

Key concepts

Exponential Growth Function

An exponential growth function is a mathematical representation of how a quantity grows over time at a rate that is proportional to its current amount. This type of function is crucial in various scientific fields, including biology for population growth, finance for compound interest, and physics for radioactive decay.

The general formula for an exponential growth function is given by \(y = a(1 + r)^t\), where:\

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• \(y\) is the future value after time \(t\)\
• \(a\) is the initial amount or the current value at \(t = 0\)\
• \(r\) is the growth rate per time period\
• \(t\) is the time period over which growth is calculated\.\

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In the context of the textbook exercise, the function \(y = 0.5e^{0.8t}\) represents exponential growth, where \(0.5\) is the initial amount, and \(e\) indicates the growth factor is based on the natural exponential constant. Understanding how to manipulate this formula is critical when modelling real-world problems that adjust compound quantities over time.

Percent Rate of Change

The percent rate of change measures how much a quantity increases or decreases relative to its previous value over a specific time period, typically expressed as a percentage. This concept is widely used to describe changes in quantities in economics, demography, finance, and other fields.

To calculate the percent rate of change, you identify the change in value, divide it by the original value, and then multiply the result by 100 to convert it to a percentage. Mathematically, if \(r\) is the growth rate, the percent rate of change is given by \(r \times 100\%\).

In the step-by-step solution, finding the percent rate of change involves adjusting the exponential function to the base \(e\) and then extracting the growth rate \(r\). Once \(r\) is determined, multiplying it by 100 provides the percentage form of the growth rate. This figure can influence decisions in scenarios like investment growth, population studies, and other applications where understanding the speed of change is vital.

Rewriting Exponential Functions

Rewriting exponential functions into a form that is easier to interpret or use for analysis is a fundamental skill in algebra and precalculus.

The exercise demonstrates how to rewrite the exponential function \(y = 0.5e^{0.8t}\) into the standardized form \(y = a(1 + r)^t\), a transformation that helps to identify the initial value \(a\) and the percent rate \(r\) more clearly. The natural exponent \(e\), approximately 2.71828, signifies continuous growth, and the rate in the given function is compounded continuously at 0.8 per time period.

Rewriting the function allows us to bridge the gap between the abstract formula and practical interpretation: identifying the starting quantity and how quickly that quantity is growing. Turning the abstract \(e\)-based expression into a practical growth model with a clear percentage rate of growth simplifies comprehension and application in varied contexts such as calculating savings account balances, predicting population sizes, or understanding the decay of substances over time.