Problem 3 - Applied Calculus

Question

Let \(W=f(t)\) represent wheat production in Argentina, \({ }^{4}\) in millions of metric tons, where \(t\) is years since 1990 . Interpret the statement \(f(12)=9\) in terms of wheat production.

Step-by-Step Solution

Answer

In 2002, Argentina's wheat production was 9 million metric tons.

Step 1: Understanding the Function

The function W=f(t) describes wheat production in Argentina in millions of metric tons, with t representing the number of years since 1990.

Step 2: Interpreting the given statement

The statement f(12)=9 tells us that when t equals 12, the function value, which is the wheat production, is 9 million metric tons.

Step 3: Identifying the Year

Since t represents years since 1990, we need to add 12 to 1990 to get the actual year. 1990 + 12 equals 2002.

Step 4: Full Interpretation

Therefore, f(12)=9 means that in the year 2002, Argentina produced 9 million metric tons of wheat.

Key Concepts

Functions

In mathematics, a function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. You can think of a function like a machine: you input something, and the machine gives you a specific output. Every function has a rule, often written as a formula, which decides what output you get with a particular input.
For instance, the function \( W = f(t) \) in this exercise tells you how much wheat Argentina produced, based on the year. Here, \( t \) represents the number of years that have passed since 1990. To find the wheat production in a particular year, you plug in the appropriate \( t \) value into the function.
When we say \( f(12) = 9 \), it means that for the input \( t = 12 \), the output is 9. In everyday terms, when 12 years have passed since 1990, Argentina produced 9 million metric tons of wheat.

Years Since Baseline

The concept 'years since baseline' is a simple way to track time starting from a specific reference year. In this problem, the baseline year is 1990. We use this starting point so we can easily convert from years since the baseline to the actual calendar year. By setting a common starting line, comparisons become straightforward and consistent.
If we know that \( t \) is the number of years since 1990, interpreting \( f(12)=9 \) becomes easier: 12 years after 1990 is the year 2002. This step is crucial as it connects the mathematical model to real-world scenarios. Understanding this helps clarify when exactly the given output (e.g., wheat levels of 9 million metric tons) occurred in known calendar years.

Unit Conversion

Unit conversion is important in this exercise because it ensures that we interpret and compare values correctly. Here, wheat production is given in millions of metric tons. This unit lets us understand the scale of production quickly.

• 1 million metric tons equals 1,000,000 metric tons.
• When we say wheat production is 9 million metric tons, it’s a shorthand for a substantial amount, simplifying complex numbers.

In the original problem, the straightforward function \( f(t) \) relates available wheat data over years using consistent units. Without converting or understanding units, we risk misinterpreting values, especially in subjects like agriculture and economics, where precise measurements are significant. Conversions, like these, put numerical data into clear perspective, allowing practical application of mathematical ideas.