Question

Discretionary Income Individuals are able to save money when their discretionary income (income after all bills are paid) exceeds their consumption. Suppose the model, \(y=-25 a^{2}+2400 a-30,700,\) represents the average discretionary income for an individual who is \(a\) years old. If the consumption model for an individual is given by \(y=160 a+7840,\) at what age is the individual able to start saving money? Round to the nearest year.

Short answer

The individual starts saving at age 23.

Step-by-step solution

Understand the Models

Identify the given models. The discretionary income model is given by \[ y = -25a^{2} + 2400a - 30,700 \] and the consumption model is given by \[ y = 160a + 7840 \].

Set Up the Equation

To find the age when the individual starts saving money, set the discretionary income model equal to the consumption model:\[-25a^{2} + 2400a - 30,700 = 160a + 7840.\]

Move All Terms to One Side

Rearrange the equation to set it to zero:\[-25a^{2} + 2400a - 160a - 30,700 - 7840 = 0.\]Then simplify the expression:\[-25a^{2} + 2240a - 38,540 = 0.\]

Solve the Quadratic Equation

Use the quadratic formula, \[a = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\], with \(a = -25\), \(b = 2240\), and \(c = -38,540\), to find the value of \(a\).\[a = \frac{{-2240 \pm \sqrt{{2240^2 - 4 (-25) (-38,540)}}}}{{2(-25)}}\]

Calculate the Discriminant

First calculate the discriminant: \[2240^2 - 4(-25)(-38,540) = 5,017,600 - 3,854,000 = 1,163,600\].

Solve for Roots

Next, find the square root: \[\sqrt{1,163,600} = 1,079.56\].Now substitute back to get the roots: \[a = \frac{-2240 + 1079.56}{-50} \approx 23.21\] and \[a = \frac{-2240 - 1079.56}{-50} \approx 66.39\].

Determine the Appropriate Age

Since we need the age at which saving starts and it must be the minimum value, round to the nearest whole year. Hence, the individual starts saving at age 23.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations. In this exercise, algebra is crucial for setting up and solving the models for discretionary income and consumption.
We start by identifying the given models:

• Discretionary income model: \( y = -25a^{2} + 2400a - 30,700 \)
• Consumption model: \( y = 160a + 7840 \)

We set these equations equal to find the age at which the individual starts saving money. This involves rearranging and simplifying the equation using algebraic principles. Finally, we solve the quadratic equation to find the value of \( a \), which represents the age.

Quadratic Formula

The quadratic formula is a method used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is given by:

\( a = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \)

In this problem, we rearranged our algebraic equation to match the quadratic form:

• \( a = -25 \)
• \( b = 2240 \)
• \( c = -38,540 \)

By substituting these values into the quadratic formula, we calculated the discriminant and found two potential ages: 23.21 and 66.39. Rounding to the nearest whole year, we determined that the individual starts saving money at age 23.

Discriminant

The discriminant is part of the quadratic formula located under the square root sign, represented as \( b^2 - 4ac \). It determines the nature of the roots of a quadratic equation:

\( \text{Discriminant} = b^2 - 4ac \)

For our equation, we calculated it as follows:

\( 2240^2 - 4(-25)(-38,540) = 5,017,600 - 3,854,000 = 1,163,600 \)

Since the discriminant is positive (1,163,600), we know that there are two real and distinct roots.
These roots were used to find the ages at which the individual's discretionary income starts exceeding their consumption. By identifying the appropriate age, we helped determine that age 23 is when saving starts.