Problem 32 - Algebra 1

Question

Use the following information. From 1894 to 1903 the number of miles of cable car track decreased by about \(10 \%\) per year. There were 302 miles of track in 1894 . Copy and complete the table. You may want to use a calculator. $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 1894 & 1896 & 1898 & 1899 & 1900 & 1901 & 1903 \\ \hline \text { Miles of track } & ? & ? & ? & ? & ? & ? & ? \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The miles of cable car track for the given years (rounded to the nearest mile) are: 1894 - 302 miles; 1896 - 244 miles; 1898 - 199 miles; 1899 - 179 miles; 1900 - 161 miles; 1901 - 145 miles; 1903 - 118 miles.

Step 1: Calculate Miles of Track in 1895

To calculate the miles of track in 1895, multiply the initial 302 miles by 90% (or 0.9). This gives you \(302 \times 0.9 = 271.8\) miles. Round this to the nearest whole number if necessary.

Step 2: Calculate Miles of Track in 1896

Using the value from the previous year, 1895, multiply it by 0.9 to find the miles of track in 1896. \(271.8 \times 0.9 = 244.62\) miles.

Step 3: Repeat for Subsequent Years

Repeat this process of multiplying by 0.9 for each subsequent year until 1903.

Step 4: Complete the table

Fill in the table with the calculated values. The table will look like this: \[\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 1894 & 1896 & 1898 & 1899 & 1900 & 1901 & 1903 \ \hline \text { Miles of track } & 302 & 244 & 199 & 179 & 161 & 145 & 118 \ \hline \end{array}\]

Key Concepts

Exponential Decay

Exponential decay describes a process where the quantity decreases at a rate proportionate to its current value. This phenomenon is widespread, seen in contexts such as radioactive decay and depreciation of assets. The problem at hand illustrates exponential decay in a real-world scenario: the gradual decrease of cable car track miles over several years.

For the cable car track miles, the 10% yearly decrease means every year, the miles of track remain only 90% of the previous year's mileage. This constant percentage translates into an exponential decay formula, expressed as \( N(t) = N_0 (1 - r)^t \), where \( N(t) \) is the amount at time \( t \), \( N_0 \) is the initial amount, and \( r \) is the decay rate. By repeatedly applying this percentage decrease, the number of miles for each year is found by using the previous year's value as the starting point for the next calculation.

Rate of Change

The rate of change is a mathematical concept that measures how a quantity changes over time. It is a crucial concept when studying movements in a quantity, whether that movement is linear or follows some other trend, such as exponential growth or decay. In our exercise, the rate of change is negative, indicating a decrease, and it's given by a percentage. This dictates how the cable car track's miles evolve yearly. To find the miles for any given year, the rate (10% decrease) is applied to the previous year's track length.

The practical implication of the rate of change here is its consistent application, highlighting the compound effect of exponential decay. It shows that while a 10% decrease might seem small for a single year, the cumulative impact over multiple years leads to a significant reduction in the total miles of the cable car track.

Rounding Numbers

Rounding numbers is a mathematical process used to make numbers easier to work with. It often simplifies figures without sacrificing much precision. In this exercise, after calculating the miles of track for each year, answers may not be whole numbers. Since we're dealing with physical miles of track, it's not practical to consider fractions of a mile.

Therefore, rounding to the nearest whole number is necessary and, typically, it's done to the digit that will make the least impact on the integrity of the data. For instance, after calculating the miles for 1895, the result is 271.8. Rounding this to the nearest whole number gives us 272 miles. You round up when the decimal is 0.5 or greater, and down if it is less; hence, 271.8 becomes 272. Rounding is crucial because it maintains the dataset's usability, especially when dealing with practical, everyday situations, such as construction where partial measures may not be feasible.