Question

The number \(V\) of varieties of suburban nondomesticated wildlife in a community is approximated by the model \(V=15 \cdot 10^{0.02 x}, \quad 0 \leq x \leq 36\) where \(x\) is the number of months since the development of the community was completed. Use this model to approximate the number of months since the development was completed when \(V=50\).

Short answer

The approximate number of months since the development was completed when \(V=50\) is roughly around 28 months.

Step-by-step solution

Write down the equation with the given value

Set the given value of \(V = 50\) into the formula \(V=15\times10^{0.02x}\) to get \(50=15\times10^{0.02x}\).

Simplify the equation

Divide both sides of the equation by 15 to isolate \(10^{0.02x}\) on one side of the equation. This gives us \(\frac{50}{15} = 10^{0.02x}\). Simplifying \(\frac{50}{15}\) will give approximately 3.33. So the simplified equation is \(3.33 = 10^{0.02x}\).

Apply natural logarithm on both sides

Take the natural logarithm (ln) on both sides of the equation which results in \(ln(3.33) = ln(10^{0.02x})\). Using the rules of logarithms, the right side can be rewritten as \(0.02x \cdot ln(10)\), giving us a simpler equation \(ln(3.33) = 0.02x \cdot ln(10)\).

Solve for \(x\)

Solve the equation for \(x\) by dividing both sides by \(0.02 \cdot ln(10)\). Simplifying \(x=\frac{ln(3.33)}{0.02 \cdot ln(10)}\) will give the number of months after which the number of wildlife varieties is 50.

Key concepts

Natural Logarithm

The natural logarithm, commonly represented as 'ln', plays a pivotal role in unraveling exponential equations. It's the inverse operation of raising Euler's number 'e' to a power. To clarify, if we have an equation like \( e^x = y \), the natural logarithm helps us to retrieve the original exponent: \( x = ln(y) \).

When we encounter an exponential term such as \( 10^{0.02x} \) in our problem, we can apply the natural logarithm to both sides to solve for the variable 'x'. This is because the natural logarithm has a special property: \( ln(a^b) = b \times ln(a) \), which enables us to pull the exponent down in front of the log, making the equation linear and solvable for 'x'.

In educational terms, understanding the concept of natural logarithm is vital when it comes to dealing with continuous growth models or decay, which are often represented by exponential functions. The power of 'ln' in simplifying complex exponents cannot be overstated, making it an essential tool in a wide range of mathematical and scientific applications.

Exponential Equation

An exponential equation is one where the variable appears in an exponent, such as \( 10^{0.02x} = 3.33 \). These equations describe situations where a quantity grows or decays at a rate proportional to its current value, which is the hallmark of exponential growth or decay.

To solve an exponential equation, one common strategy is to express both sides of the equation with the same base. However, when matching bases isn't straightforward, using logarithms, specifically the natural logarithm, becomes a super-effective method.

In our exercise, once we isolate the exponential term, taking the natural logarithm of both sides allows us to bring the exponent down. This simplifies the equation, making it possible to solve for the unknown variable. Exponential equations like this are crucial in modeling real-world phenomena such as population growth, radioactive decay, and inflation in economics.

Mathematical Modeling

Mathematical modeling is a method of creating equations or formulas that represent real-world phenomena. In our example, the model \( V=15 \times 10^{0.02x} \) describes the relationship between time and the variety of wildlife in a suburban community.

The process of mathematical modeling typically includes observing patterns or trends, formulating a model, and validating it through additional observations or experiments. In this context, models not only help in understanding the current state but also in predicting future scenarios.

The exponential model in the exercise captures the essence of how wildlife variety may increase over time following a specific growth rate. By integrating mathematical concepts such as exponential growth with natural settings, this type of modeling proves invaluable in fields such as biology, environmental science, and urban planning. It's a powerful example of how abstract mathematical concepts are applied to make tangible predictions about the world around us.