Question

The daily demand \(D\) (in thousands of barrels) for refined oil in the United States from 1995 to 2005 can be modeled by \(D=276.4 t+16,656, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995. (a) Use the model to find the year in which the demand for U.S. oil exceeded 18 million barrels a day. (b) Use the model to predict the year in which the demand for U.S. oil will exceed 22 million barrels a day.

Short answer

Based on the model, the U.S. oil demand exceeded 18 million barrels a day in 1995 and it is predicted to exceed 22 million barrels a day in 2010.

Step-by-step solution

Find the year when the U.S. oil demand exceeded 18 million barrels a day

Substitute \(D = 18000\) in the equation \(D=276.4 t+16,656\). We get \(18000=276.4t + 16656\). Solve this equation for \(t\) to get \(t= (18000-16656)/276.4 \approx 4.86\). Adding 1990, we get approximately the year 1994.86, which rounds up to 1995. Thus, the demand exceeded 18 million barrels a day in 1995.

Predict the year when the U.S. oil demand will exceed 22 million barrels a day

Substitute \(D = 22000\) in the equation \(D=276.4 t+16,656\). We get \(22000=276.4t + 16656\). Solve this equation for \(t\) to get \(t= (22000-16656)/276.4 \approx 19.34\). Adding 1990, we get approximately the year 2009.34, which rounds up to 2010. Thus, the model predicts that the demand will exceed 22 million barrels a day in 2010.

Key concepts

Linear Equations

Understanding linear equations is crucial when modeling real-world scenarios, like predicting oil demand. A linear equation represents a straight line when plotted on a graph and has the general form of \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. In the context of the oil demand prediction model, the linear equation is given by \( D = 276.4t + 16,656 \), linking oil demand \( D \) with the time \( t \) in years.

In the exercise solution, to find when oil demand exceeded a certain threshold, the known demand values are substituted into the D variable—this is effectively finding the x-intercept of the line at a given y value (demand). Solving for \( t \) gives us the specific point in time at which the threshold is crossed. This task unveils linear equations' fundamental property: they provide a straight-forward, proportionate relationship between two variables.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world situation to predict or analyze future events. The oil demand prediction model \( D = 276.4t + 16,656 \) is a perfect example, where \( t \) and \( D \) correspond to time and demand respectively.

The model simplifies complex realities into an algebraic equation, allowing us to visualize and estimate future demand based on past and present trends. For instance, by setting the equation up with a certain demand as a threshold, you can estimate the year when the demand will reach that threshold. It is important to note that while mathematical models can be incredibly useful, they also rely on the assumption that current trends will continue and do not account for unforeseen future changes in technology, policy or consumer behavior that might affect oil demand.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific mathematical quantity, like the demand for oil in our example. They are the building blocks of algebraic equations. In the given oil demand prediction model, \( 276.4t + 16,656 \) is the algebraic expression for the demand \( D \) based on time \( t \) as the variable.

When working with algebraic expressions within an equation, one often performs operations such as substitution—as seen when we find the value of \( t \) that meets a certain demand in the problem. Proper manipulation of these expressions is essential to solve for an unknown variable and gives significant insight into the relationships between different elements of a problem or scenario.