The exponential growth of the deer population in...

The exponential growth of the deer population in Massachusetts can be approximated using the model $$f(x)=50,000(1+0.06)^{x}$$ where $50,000$ is the initial deer population and 0.06 is the rate of growth. $f(x)$ is the total population after $x$ years have passed. Find each value to the nearest thousand. (a) Predict the total population after 4 yr. (b) If the initial population was $30,000$ and the growth rate was $0.12,$ how many deer would be present after 3 yr? (c) How many additional deer can we expect in 5 yr if the initial population is $45,000$ and the current growth rate is $0.08 ?$

The exponential growth of the deer population in Massachusetts can be approximated using the model
$$f(x)=50,000(1+0.06)^{x}$$
where $50,000$ is the initial deer population and 0.06 is the rate of growth. $f(x)$ is the total population after $x$ years have passed. Find each value to the nearest thousand.
(a) Predict the total population after 4 yr.
(b) If the initial population was $30,000$ and the growth rate was $0.12,$ how many deer would be present after 3 yr?
(c) How many additional deer can we expect in 5 yr if the initial population is $45,000$ and the current growth rate is $0.08 ?$

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Key Concepts

Rounding Approximations

Rounding approximations involve adjusting calculated numerical results to a specified degree of accuracy – in this case, to the nearest thousand. This process simplifies the results, making them easier to interpret while still reflecting an estimate of the actual value.

Exponential Functions

Exponential functions are used to model scenarios where a quantity increases (or decreases) by a constant percentage per time period. In this context, they take the form f(x) = initial_value * (1 + growth_rate)^x, where the exponent x represents the number of time periods.

Initial Value

The initial value represents the starting amount of the quantity being modeled. It sets the baseline from which exponential growth begins and is crucial for calculating future values using the exponential model.

Growth Rate

The growth rate is the rate at which the quantity increases per time period, expressed as a decimal. It determines the percentage change applied to the current amount in each period, driving the exponential growth.

Time Variable

The time variable represents the number of time periods (such as years) that pass. It serves as the exponent in the exponential function, determining how many times the growth factor is applied.

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