Question

Domestic Demand The U.S. domestic demand \(D\) (in millions of barrels) for refined oil products from 1995 to 2005 can be modeled by \(D=100.9708 t+6083.999,5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. The population \(P\) (in millions) of the United States from 1995 to 2005 can be modeled by \(P=3.0195 t+251.817, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. (Sources: U.S. Energy Information Administration and the U.S. Census Bureau) (a) Construct a rational function \(B\) to describe the per capita demand for refined oil products. (b) Use a graphing utility to graph the rational function \(B\). (c) Use the model to predict the per capita demand for refined oil products in \(2010 .\)

Short answer

The per capita demand for refined oil products in 2010 is predicted by the model to be around 364.7 million barrels

Step-by-step solution

Converting time to \(t\) formula

First it should be noticed that the time \(t\) for 2010 would be 20 because in the problem it was defined that \(t=5\) for the year 1995, this means the increment is 1 per each passed year.

Construct the rational function

The rational function \(B(t)\) that describes the per capita demand for refined oil products is given by the division of \(D(t)\) by \(P(t)\), i.e., \(B(t) = D(t) / P(t)\). This means every person's share of oil demand.

Predict per capita demand for 2010

Once we have the model \(B(t)\), we can use it to predict the per capita demand for refined oil products in 2010. Doing this we get \(B(t) = \frac{{100.9708 * 20 + 6083.999}}{{3.0195 * 20+ 251.817}}\)

Key concepts

Per Capita Demand

The term 'per capita demand' is a way to describe the average quantity of a particular resource each person in a population uses. It's calculated by dividing the total demand for a resource by the total population. This concept is particularly useful for understanding how resource usage scales with population changes.
In our example, the per capita demand for refined oil products in the U.S. is calculated by using a rational function. A rational function can be thought of as a ratio of two expressions, representing different measurements. Here, it represents the total demand for refined oil products divided by the total population.
To construct this rational function for the per capita demand, we use the model:

• Total Demand: \( D(t) = 100.9708t + 6083.999 \)
• Total Population: \( P(t) = 3.0195t + 251.817 \)

The rational function is then given by: \( B(t) = \frac{D(t)}{P(t)} \). This function allows us to compute how much each person, on average, demands refined oil over time.

Refined Oil Products

Refined oil products are essential components of modern economies, derived from crude oil through processes like refining and distillation. These products include gasoline, diesel, jet fuel, and others that power transportation networks and industries.
The demand for refined oil products can be modeled mathematically, as we saw with the function \( D(t) \). By using such models, we can project future demands and assess how factors like technological advances, policy shifts, or changes in consumer behavior might affect consumption.
In this context, understanding both total and per capita demand helps planners and policymakers make informed decisions about energy, resource management, and infrastructural investments. It highlights the dependency on oil and provides a basis for exploring alternative energy sources to meet long-term sustainability goals.

Population Modeling

Population modeling uses mathematical functions to describe how populations change over time. These models help us understand growth trends and predict future populations.
In our example, the population of the U.S. from 1995 to 2005 is represented by the linear equation \( P(t) = 3.0195t + 251.817 \). This model indicates that the population grows steadily each year, at a rate dictated by the coefficient of \( t \).
Population models are crucial for resource planning. As populations increase, the demand for resources such as refined oil products also rises. Policymakers and economists use these models to ensure that supply meets demand, aiding in infrastructure planning and environmental management.

• Support decisions on resource allocation
• Predict future economic and social needs
• Develop policies for sustainable development

By understanding population modeling, stakeholders can better anticipate and prepare for demographic changes impacting demand and supply dynamics.