Problem 87 - Elementary Algebra

Question

The value in dollars of a new car is modeled by the function V(t)=125t2?3,000t+22,000, where t represents the number of years since it was purchased. Determine the age of the car when its value is at a minimum.

Step-by-Step Solution

Answer

The car's value is minimum at 12 years.

Step 1: Understand the problem

The function given is a quadratic equation \( V(t) = 125t^2 - 3000t + 22000 \), where \( t \) represents the age of the car in years. We need to find the age \( t \) when the value \( V(t) \) is at its minimum.

Step 2: Identify the type of function

The given function \( V(t) = 125t^2 - 3000t + 22000 \) is a quadratic function in the form \( at^2 + bt + c \), where \( a = 125 \), \( b = -3000 \), and \( c = 22000 \). Quadratic functions have a parabolic shape, and since \( a > 0 \), the parabola opens upwards, indicating there is a minimum value at the vertex.

Step 3: Use the vertex formula

The vertex of a quadratic function \( at^2 + bt + c \) occurs at \( t = -\frac{b}{2a} \). In our case, \( a = 125 \) and \( b = -3000 \). Substitute these values into the vertex formula to find \( t \).

Step 4: Calculate the age at minimum value

Calculate \( t \) using the vertex formula: \[ t = -\frac{-3000}{2 \times 125} = \frac{3000}{250} = 12 \] Therefore, the car's value reaches its minimum at \( t = 12 \) years.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. It is the graphical representation of a quadratic function. The axis of symmetry is a vertical line that passes through the vertex, which is the point where the curve opens.
The direction of the parabola—whether it opens up or down—is determined by the coefficient of the squared term:

• If the coefficient is positive, the parabola opens upwards, indicating that it has a minimum point.
• If it is negative, the parabola opens downwards, indicating a maximum point.

This specific curve is symmetric, meaning if you fold it along its line of symmetry (the axis of symmetry), both halves match perfectly.

Vertex Formula

The vertex formula is a key tool in finding the turning point, or vertex, of a quadratic function. This point can represent either a maximum or minimum value of the function, depending on the parabola's direction.The vertex of a quadratic function in the form \( ax^2 + bx + c \) is calculated using the formula:
\[t = -\frac{b}{2a}\]Here:

• \(a\) is the coefficient of \(t^2\)
• \(b\) is the coefficient of \(t\)
• \(c\) is the constant term

Using this formula helps you quickly determine the critical point of the parabola, which is crucial for solving problems related to quadratic functions.

Minimum Value

The minimum value of a quadratic function occurs at its vertex when the parabola opens upwards. The shape of the graph will bottoms out at this point, making it the lowest value that the function attains.
The process to find the minimum value involves two main steps:

• Identifying or computing the vertex, which provides the \(t\)-value when it happens.
• Substituting this \(t\)-value back into the original quadratic function to compute the actual minimum value.

In real-life applications, like the depreciation of a car's value, understanding this minimum point can provide valuable insights into the optimal timeframes or values.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically expressed as \( ax^2 + bx + c = 0 \). It's a foundational concept in algebra, used to model various real-world scenarios.
The solutions to a quadratic equation are called the roots, and they can be found using several methods:

• Factoring
• Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

Quadratic equations can have real or complex roots, and understanding how to interpret their graphs and solutions is critical in fields ranging from physics to economics.