Problem 15 - Applied Calculus

Question

Table \(1.11\) shows world bicycle production. \(^{28}\) (a) Find the change in bicycle production between 1950 and 2000 . Give units. (b) Find the average rate of change in bicycle production between 1950 and 2000. Give units and interpret your answer in terms of bicycle production. $$ \begin{array}{l} \text { Table 1.11 World bicycle production, in millions }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Bicycles } & 11 & 20 & 36 & 62 & 92 & 101 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

(a) 90 million bicycles; (b) 1.8 million bicycles per year, indicating an average yearly increase in production.

Step 1: Identify the Values

To find the change in bicycle production between 1950 and 2000, we first need to identify the production values from these years. From the table, we see that in 1950 the production was 11 million, and in 2000, it was 101 million.

Step 2: Calculate the Change in Production

The change in production from 1950 to 2000 can be calculated by subtracting the production in 1950 from the production in 2000. So, \[\text{Change in Production} = 101 - 11 = 90 \text{ million bicycles}.\]

Step 3: Determine the Time Interval

The time interval between 1950 and 2000 is the difference in years: \[2000 - 1950 = 50 \text{ years}.\]

Step 4: Calculate the Average Rate of Change

To find the average rate of change, divide the change in production by the time interval. This gives us the average change in production per year:\[\text{Average Rate of Change} = \frac{90 \text{ million bicycles}}{50 \text{ years}} = 1.8 \text{ million bicycles per year}.\]

Step 5: Interpret the Average Rate of Change

The average rate of change of 1.8 million bicycles per year means that, on average, the world bicycle production increased by 1.8 million bicycles every year between 1950 and 2000.

Key Concepts

Bicycle Production

Bicycle production has been a key indicator of mobility and economic trends around the world. To understand how bicycle production has changed over time, it's helpful to examine yearly production figures. For instance, in the year 1950, the world produced 11 million bicycles. By 2000, this number grew significantly to 101 million bicycles. These figures help us track how the demand and the manufacturing capability have evolved over the years. Each increase could reflect advancements in manufacturing technology, economic growth, or increased demand for bicycles as a mode of transportation. Analyzing these numbers not only helps in understanding past trends but also in planning for future infrastructure and production goals.

Units Interpretation

Interpreting units is essential to understanding what the numbers mean in real-world terms. In the bicycle production example, the units are provided in millions of bicycles. This scale helps comprehend the large volume involved. For a change calculation, like the jump from 11 million bicycles in 1950 to 101 million in 2000, the difference is expressed as 90 million bicycles.

• When discussing "change" over a 50-year period, we emphasize total production increases, with 90 million more bicycles produced in 2000 compared to 1950.
• Understanding the "average rate of change" involves dividing this total change by the number of years (50 years), resulting in units of "million bicycles per year." This tells us how much production increased annually on average.

Knowing the units helps in grasping the scale of change and comparing it with other industries or periods. On a broader scale, such insights can aid policy making, especially in transportation and environmental planning.

Step-by-Step Solution

A thorough step-by-step solution aids in understanding complex problems by breaking them into manageable parts. Here's how you can apply this approach to the problem of bicycle production: 1. **Identify the Values**: Begin by pinpointing relevant data from provided resources. For example, note that in 1950 production was 11 million bicycles and increased to 101 million by 2000.
2. **Calculate the Change**: Find the change in production by subtraction: We have 101 - 11 = 90 million bicycles.
3. **Determine the Time Interval**: Identify the duration over which this change occurred (2000 - 1950 = 50 years).
4. **Average Rate of Change Calculation**: Divide the total change by the time duration to determine how much the production changes per year: We get 1.8 million bicycles per year.
5. **Interpret the Results**: This means an increase of 1.8 million bicycles annually on average during this period, which reflects ongoing trends and potential shifts in societal transportation needs.
This methodical approach not only clarifies questions but also fosters confidence in tackling similar future problems.