Problem 55 - College Algebra

Question

The formula \(A=18.9 e^{0.0055 t}\) models the population of New York State, \(A\), in millions, \(t\) years after 2000 . a. What was the population of New York in \(2000 ?\) b. When will the population of New York reach 19.6 million?

Step-by-Step Solution

Answer

The population of New York in 2000 was 18.9 million. The population of New York State will reach 19.6 million approximately \(t\) years after 2000 where \(t\) is found by numeric computation on the final formula from part (b).

Step 1: Find the population in 2000

To find the population in 2000, we set \(t=0\), as that is the year from which our time is calculated. Substituting into the equation, we get \(A = 18.9 e^{0.0055*0}\). Recalling that any number to the power of 0 equals 1, our equation then becomes \(A = 18.9*\). Hence, in the year 2000, the population was 18.9 million.

Step 2: Determine when the population will reach 19.6 million

This step requires setting \(A = 19.6\) million and solving the equation for \(t\). This gives us 19.6 = 18.9 e^{0.0055 t}. We can then isolate the term with the variable by dividing both sides by 18.9, which gives us \(e^{0.0055 t} = 19.6 / 18.9\). To solve for \(t\), we take the natural logarithm(ln) of both sides so as to reduce the exponent, which gives us \(0.0055 t = ln(19.6 / 18.9)\). Lastly, we solve for \(t\), which gives \(t = ln(19.6 / 18.9) / 0.0055\), which can be evaluated approximately (using a calculator with a natural logarithm function) to obtain \(t\).

Step 3: Check the validity of the result

The calculated value for \(t\) should be positive, since it represents the number of years after 2000. Also for part (a), the result should match with the given initial population.

Key Concepts

Population Modeling

Population modeling is a mathematical way to describe how a population changes over time. We use equations to project future growth or decline in populations based on certain assumptions, like birth rates, death rates, and migration patterns. In this example, we use an exponential growth model to predict the population of New York State.The equation given is:\[A = 18.9 \, e^{0.0055 \, t}\]where:

• \(A\) is the population (in millions).
• \(t\) is the time in years after the year 2000.
• 18.9 represents the initial population size in millions at year 2000.
• 0.0055 is the growth rate per year.

By using this model, we can predict the population at some future time \(t\). This is especially useful for planning resource allocation, urban development, and infrastructure requirements.

Natural Logarithm

The natural logarithm, denoted as \(ln\), is a special logarithm that uses the base \(e\). The number \(e\) is an irrational and transcendental number approximately equal to 2.71828.Natural logarithms are inverse operations to exponentials. This means that the natural logarithm of \(e^x\) is simply \(x\). In mathematical terms:\[ln(e^x) = x\]In the context of population models, natural logarithms help us solve for time \(t\) in equations where population is modeled using exponential growth. They simplify the process of isolating variables when they are in the exponent, allowing us to find exact numerical values.

Solving Exponential Equations

To solve exponential equations like \(19.6 = 18.9 \, e^{0.0055 \, t}\), it is critical to isolate the exponential term before applying the natural logarithm to both sides of the equation.Here's how you do it:

• First, divide both sides of the equation by 18.9 to get the exponential part on its own: \[e^{0.0055 \, t} = \frac{19.6}{18.9}\]
• Next, apply the natural logarithm to both sides to remove the exponential function:\[ln(e^{0.0055 \, t}) = ln\left( \frac{19.6}{18.9} \right)\]This simplifies to:\[0.0055 \, t = ln\left( \frac{19.6}{18.9} \right)\]
• Lastly, solve for \(t\) by dividing by 0.0055:\[t = \frac{ln\left( \frac{19.6}{18.9} \right)}{0.0055}\]

Now you can use a calculator to determine the exact value of \(t\), which tells you how many years after 2000 the population reaches 19.6 million.

Initial Value Problems

An initial value problem (IVP) is a type of problem that involves finding a function or a number that satisfies a given mathematical model at a starting point, usually time zero. In terms of population modeling, the initial value represents the population size at the base year, which in this case is the year 2000.The problem given starts us off with the equation:\[A = 18.9 \, e^{0.0055 \, t}\]Here, the initial value \(A_{0}\) is 18.9, which is the population of New York State at the start.For part (a) of our exercise, we confirm this by substituting \(t = 0\):

• \(A = 18.9 \, e^{0.0055 \, \times \, 0} = 18.9 \, e^0 = 18.9\)

This shows that the initial model accurately represents a population of 18.9 million in the year 2000, confirming the correctness of our exponential growth model setup.