Problem 106 - College Algebra and Calculus: An Applied Approach

Question

The daily amount \(I\) (in thousands of barrels) of crude oil imported to the United States from 1995 to 2005 can be modeled by \(I=428.2 t+6976, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995. (a) Use the model to find the year in which the amount of crude oil imported to the United States exceeded 12 million barrels a day. (b) Use the model to predict the year in which the amount of oil imported to the United States will exceed 14 million barrels a day.

Step-by-Step Solution

Answer

The year the U.S imported more than 12 million barrels a day is 1999. While, it is predicted that U.S will import more than 14 million barrels a day by 2005.

Step 1: Convert the barrel amounts to the same units

Given amounts in questions are in millions of barrels, but equation outputs in thousands. Therefore, firstly convert the given numbers in millions to thousands. For instance, 12 million barrels will be 12,000 thousands of barrels and 14 million barrels will be 14,000 thousands of barrels.

Step 2: Find the year when U.S imported more than 12 million barrels a day

Plug the number 12,000 into the formula \(I=428.2 t+6976\) for \(I\) and then solve for \(t\). The resulting \(t\) will be number of years after 1995 so to get the actual year, add 1995 to your result.

Step 3: Predict the year when U.S will import more than 14 million barrels a day

Similar to step 2, plug 14,000 into the formula \(I=428.2 t+6976\) for \(I\) and solve for \(t\). The resulting \(t\) will be years after 1995 so add 1995 to your result to get the actual year.

Step 4: Verify your answers

Make sure your results are within the domain \(5 \leq t \leq 15\), as the model is valid only for these years.

Key Concepts

Crude Oil Import Prediction

Predicting the import of crude oil involves understanding a mathematical model that represents the trend over a period of time. In this exercise, the model is given by the equation \(I=428.2t+6976\), where \(I\) is the daily import of crude oil in thousands of barrels, and \(t\) is the year, with \(t=5\) corresponding to 1995.
Forecasting such imports helps governments and industries plan for future needs, maintain energy security, and manage resources efficiently. Understanding the point at which imports exceed a certain threshold, like 12 million or 14 million barrels a day, requires analyzing this model to find the specific year when these levels are anticipated.
If you're tasked with predicting this, you'll use given numbers and solve the equation to pinpoint the year.

Linear Equations

Linear equations model relationships with a constant rate of change, where the graph of the relationship is a straight line. The equation \(I=428.2t+6976\) is a linear equation that helps us determine the amount of crude oil imported over specific years from 1995 to 2005.
In the equation, \(428.2t\) represents the change per year, and \(6976\) is the starting point, or intercept, meaning the basic level of imports when \(t=0\).

• The slope, \(428.2\), shows the growth in daily imports in thousands of barrels per year.
• The intercept, \(6976\), represents the baseline imports in the model's first year, translated from \(t=0\).

Solving linear equations like this enables us to make informed predictions, provided our values for \(t\) stay within the model's valid range of \(5 \leq t \leq 15\).
By setting \(I\) to different values, we solve for \(t\) to find specific years, highlighting linear equations' utility in problem-solving.

Unit Conversion

Understanding and applying unit conversions is essential when working with mathematical models, especially those describing real-world situations. In this instance, the model gives \(I\) as thousands of barrels, while the problem demands solutions in millions of barrels. For accurate predictions, you must convert units accordingly.
To convert millions of barrels into thousands, multiply by 1,000. For example:

• 12 million barrels becomes 12,000 thousand barrels.
• 14 million barrels becomes 14,000 thousand barrels.

This step is crucial to ensure that calculations align with the model, allowing for straightforward and correct determination of significant values like the year imports exceed given levels. By mastering unit conversions, you enhance your problem-solving skills in various mathematical modeling tasks.

Problem Solving

Problem solving in mathematical modeling involves applying logical steps to interpret and solve an equation based on the information given. Working through the crude oil import scenario, we use a sequence of straightforward yet essential steps.
Here's how problem solving is structured for this model:

• Identify the information given and the question asked.
• Make necessary conversions to adapt the information within the model's context.
• Substitute values into the equation to solve for unknowns.
• Check the outcome to ensure it lies within the acceptable range of the model.

By methodically following these steps, you ensure accuracy in your results and refine your approach to tackling mathematical problems. This structured approach is applicable across various scenarios, enhancing your analytical and logical skills in real-world practices.