Question

At the corner Shell station, the revenue \(R\) varies directly with the number \(g\) of gallons of gasoline sold. If the revenue is 34.08 when the number of gallons sold is \(12,\) find a linear equation that relates revenue \(R\) to the number \(g\) of gallons of gasoline sold. Then find the revenue \(R\) when the number of gallons of gasoline sold is \(10.5 .\)

Short answer

The linear equation is \( R = 2.84g \). The revenue when selling 10.5 gallons is 29.82.

Step-by-step solution

Understanding Direct Variation

When revenue, denoted as R, varies directly with the number of gallons of gasoline sold, denoted as g, it means that we can express this relationship as R = kg, where k is a constant.

Find the Constant of Proportionality

Given that R is \(34.08\) when g is 12, substitute these values into the equation R = kg to find k: \[ 34.08 = k \times 12 \]. Solving for k gives \[ k = \frac{34.08}{12} = 2.84 \].

Write the Linear Equation

With the value of k found, the linear equation relating revenue R to the number of gallons g is \[ R = 2.84g \].

Calculate Revenue for 10.5 Gallons

Substitute g = \(10.5\) into the equation \[ R = 2.84g \]: \[ R = 2.84 \times 10.5 = 29.82 \].

Key concepts

linear equations

Linear equations are a fundamental concept in algebra. They represent a straight line when graphed on a coordinate plane. The general form of a linear equation is written as: \[ y = mx + b \] Here, 'y' is the dependent variable, 'x' is the independent variable, 'm' is the slope of the line, and 'b' is the y-intercept. In the context of the problem, the equation \[ R = 2.84g \] is a linear equation because it describes a straight-line relationship between revenue (\(R\)) and gallons sold (\(g\)). The slope 'm' in this case is the proportionality constant (2.84), and there is no y-intercept (b=0) since the line passes through the origin, meaning if no gallons are sold, revenue is zero.

proportionality constant

The proportionality constant is an important part of direct variation in linear relationships. It represents the rate at which the dependent variable changes with respect to the independent variable. In our example: Given the equation \(R = 2.84g\), the constant of proportionality (k) is 2.84. This constant signifies that for every gallon of gasoline sold, the revenue increases by \(2.84\) dollars. To find this constant, we used the values from the problem: revenue \(R = 34.08\) and gallons \(g = 12\). Substituting these into the equation \(R = kg\) provides: \[ 34.08 = k \times 12 \] Solving for 'k', we get: \[ k = \frac{34.08}{12} = 2.84 \]

revenue calculation

In problems involving direct variation, calculating revenue can be straightforward. Revenue (R) depends linearly on the number of gallons sold (g). With the proportionality constant (k) found, we can easily compute the revenue for any number of gallons. For example, to find the revenue when 10.5 gallons are sold, we use the equation: \[ R = 2.84g \] Substituting \(g = 10.5\): \[ R = 2.84 \times 10.5 = 29.82 \] Thus, the revenue when 10.5 gallons are sold is \(29.82\) dollars.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or set of values. Understanding how to manipulate these expressions is critical in solving algebra problems. In our direct variation scenario, the algebraic expression \(R = 2.84g\) tells us how revenue changes with the number of gallons sold. By substituting given values into the expression, we can find the unknown variable. For instance, substituting \(g = 12\) helped us find the constant of proportionality: \[ 34.08 = k \times 12 \] Solving it, we found \( k = 2.84 \), and then used this to form the expression \( R = 2.84g \).

word problems in algebra

Word problems in algebra require translating a written scenario into mathematical equations. This process often follows a specific methodical approach:

Read the problem carefully

Identify and define the variables involved

Write down the equations

Solve for the unknowns

Interpret the solution in the context of the problem

In our example, the problem provides the relationship between revenue and gallons of gasoline, leading us to form the equation \(R = 2.84g\). Understanding the steps and knowing how to apply algebraic techniques helps solve such problems effectively. By practicing this approach, students can better tackle similar word problems in algebra.