Problem 57 - College Algebra

Question

The quadratic function $$ f(x)=-0.018 x^{2}+1.93 x-25.34 $$ describes the miles per gallon, \(f(x),\) of a Ford Taurus driven at \(x\) miles per hour. Suppose that you own a Ford Taurus. Describe how you can use this function to save money.

Step-by-Step Solution

Answer

To save money on gas, drive the car at a speed of approximately 53.61 miles/hour. At this speed, the car gives its maximum gas mileage of 25.34 miles per gallon.

Step 1: Identify the Quadratic Function

The quadratic function is given as \(f(x) = -0.018x^2 + 1.93x - 25.34\). This is in the form \(f(x) = ax^2 + bx + c\), where 'a' is -0.018, 'b' is 1.93, and 'c' is -25.34.

Step 2: Find the Vertex of the Function

The vertex of the parabola described by the function \(f(x) = ax^2 + bx + c\) is given by the point \((-b/2a, f(-b/2a))\). So, determine the 'x' value of the vertex using the equation \(-b/(2a)\). By substituting 'b' and 'a' into the equation, the 'x' value becomes \(-1.93/(2 * -0.018) = 53.61\). Now, find the 'y' value by plugging 'x' back into the function: \(f(53.61) = -0.018(53.61)^2 + 1.93(53.61) - 25.34 = 25.34\). Thus, the vertex of the parabola is \((53.61, 25.34)\), which is the maximum point of the function.

Step 3: Interpretation of Result

The maximum point \((53.61, 25.34)\) implies that the maximum gas mileage of 25.34 miles per gallon is achieved when the car is driven at a speed of 53.61 miles/hour. Thus, to save money on gas, consider driving at approximately 53.61 miles/hour, as this speed gives the highest miles per gallon.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when dealing with quadratic functions, such as the given function for fuel efficiency. In general terms, the vertex of a parabola represents the highest or lowest point on the graph, depending on the direction the parabola opens.

For a quadratic function in the form of \( f(x) = ax^2 + bx + c \), where 'a', 'b', and 'c' are constants, the vertex can be calculated using the formula \( x = -\frac{b}{2a} \). Once the 'x' value is found, it can be plugged back into the function to find the corresponding 'y' value. In the case of the fuel efficiency function, we calculate the 'x' value as \( -\frac{1.93}{2(-0.018)} \) which results in the speed that maximizes the car's fuel efficiency.

The 'y' value is the actual fuel efficiency at that speed, which allows car owners to optimize their fuel usage by driving at the speed corresponding to the vertex of the parabola.

Maximizing Fuel Efficiency

Fuel efficiency is a critical aspect of cost-saving in vehicle maintenance. The given quadratic function provides a model for maximizing the fuel efficiency of a Ford Taurus. By finding the vertex of the parabola, we can determine the most efficient speed to drive to achieve maximum miles per gallon (mpg).

In practical terms, once the optimal speed is identified, maintaining your speed around this point during travel could lead to significant cost savings on fuel. However, it's also essential to consider that driving conditions, such as traffic and road types, may affect the ability to maintain this optimal speed.

Applications of Quadratic Functions

Quadratic functions are utilized in various real-world scenarios, not just for mathematical curiosity. The case of optimizing fuel efficiency represents just one practical application of these functions. Quadratic relations can model phenomena such as projectile motion, market economics (supply and demand curves), and even biology (growth rates).

Moreover, recognizing the shape of a quadratic graph helps in understanding how changes to the function's coefficients—'a', 'b', and 'c'—affect the trajectory and orientation of the curve. This insight is invaluable for professionals across many fields who use these functions to predict, analyze, and optimize outcomes.