Problem 43 - Precalculus

Question

Explain the mistake that is made. The city of Orlando, Florida, has a population that is growing at \(7 \%\) a year, compounding continuously. If there were 1.1 million people in greater Orlando in \(2006,\) approximately how many people will there be in \(2016 ?\) Apply the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of people. Solution: Use the population growth model. \(N=N_{0} e^{r t}\). Let \(N_{0}=1.1, r=7,\) and \(t=10 . \quad N=1.1 e^{(7)(10)}\). Approximate with a calculator. \(2.8 \times 10^{30}\). This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was substituting the growth rate as 7 instead of 0.07.

Step 1: Identify the Given Values

We are given the following values: the initial population \(N_0 = 1.1\) million, the growth rate \(r = 7\% = 0.07\) (as a decimal), and the time period \(t = 10\) years (from 2006 to 2016).

Step 2: Analyze the Formula Substitution

The solution provided substitutes these values into the formula as follows: \(N = 1.1 e^{(7)(10)}\), making a mistake in substituting the growth rate \(r\) as \(7\) instead of \(0.07\).

Step 3: Correct the Growth Rate Substitution

Correct the substitution of the growth rate into the formula. Using the correct decimal, the equation should be \(N = 1.1 e^{0.07 imes 10}\).

Step 4: Recalculate Using Correct Values

Now calculate using the correct formula: \(N = 1.1 e^{0.7}\). Use a calculator to find \(e^{0.7} \approx 2.01375\). Thus, \(N \approx 1.1 \times 2.01375 = 2.215125\) million.

Step 5: Conclusion and Error Identification

The error was the use of the incorrect growth rate in the exponential function. By using \(7\) instead of \(0.07\), the estimated population became unrealistically large.

Key Concepts

Continuous Compounding

Continuous compounding is an essential concept in both finance and population dynamics. When growth compounds continuously, it means that the quantity is not just increasing at set intervals but is in a constant state of growth.
This means every moment new growth builds on the previous growth. In practice, it is modeled mathematically using the exponential function.

• The basic formula used is: \[ N = N_0 e^{rt} \]Here, \(N\) is the final amount, \(N_0\) is the initial quantity, \(r\) is the continuous growth rate, and \(t\) is time.
• The number \(e\) is the base of the natural logarithm, approximately equal to 2.71828. This mathematical constant plays a key role in continuous growth scenarios.

Continuous compounding leads to faster growth compared to discrete growth.
In the context of the exercise, incorporating it accurately allows for the correct calculation of the population over a given time period.

Population Growth Model

A population growth model is a mathematical representation used to predict how a population will grow over time. This model takes into account how populations operate under natural laws.
In our case, the model uses exponential growth, reflecting realistic conditions seen in nature where resources influence growth.

• The exponential formula \[ N = N_0 e^{rt} \] helps to calculate future population sizes.
• In this equation, \(N_0\) represents the initial population at the start of observation, \(r\) represents the growth rate per unit of time, and \(t\) is the length of the time period over which growth is supplied.

The proper use of this model requires precise insertion of values, especially the growth rate \(r\).
An error in these inputs can lead to unrealistic outcomes, as seen when the problem used 7 instead of 0.07, leading to unmanageable growth predictions.

Common Errors in Math Calculations

Math calculations can often go wrong due to simple mistakes, especially when dealing with formulas that require precise inputs.
One frequent mistake involves incorrect substitutions of values into a formula, leading to wrong results.

• Using the wrong unit or format for a number, such as substituting percentage growth rate directly rather than converting it to a decimal.
• In the current exercise, substituting \(r = 7\) instead of \(r = 0.07\) led to an exponential calculation way bigger than expected, wildly exaggerating the population size.
• Miscalculations or misinterpretations of multiplication orders further compound these errors, especially in complex equations like the exponential growth model.

Double-checking each step is vital to avoid these mistakes.
Using calculators correctly, validating the logic of your substitutions, and making sure all units are in their proper formats can save time and lead to more accurate solutions.