For the following exercises, consider this scenario: For...

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.) Which forest had a greater number of trees initially? By how many?

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Which forest had a greater number of trees initially? By how many?

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Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to compute its output. In many applied problems, evaluating the function at time zero reveals the starting quantity, which serves as a reference point for further analysis.

Comparison of Quantities

Comparing quantities involves computing differences between values to determine relative magnitude. This concept is fundamental when assessing which entity starts with a greater amount, such as determining which population is larger initially.

Exponential Functions

Exponential functions model processes where a quantity increases or decreases at a rate proportional to its current value. They are essential in understanding phenomena such as population growth, where the base of the exponent represents the growth factor over equal time intervals.

Initial Value

The initial value of an exponential function is the constant term that represents the quantity at time zero, before any growth or decay has occurred. This value is critical for comparing baseline figures in various contexts.

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