Question

Student Working A study found that a student's GPA, \(g\), is related to the number of hours worked each week, \(h,\) by the equation \(g=-0.0006 h^{2}+0.015 h+3.04 .\) Estimate the number of hours worked each week for a student with a GPA of \(2.97 .\) Round to the nearest whole hour.

Short answer

30 hours

Step-by-step solution

- Substitute GPA into the Equation

Start by substituting the given GPA into the equation. We set \(g = 2.97\) in the equation \(g = -0.0006h^2 + 0.015h + 3.04\)

- Set Up the Equation to Solve for Hours Worked

The equation becomes: \(2.97 = -0.0006h^2 + 0.015h + 3.04\). Now, rearrange it to get a standard form quadratic equation: \(-0.0006h^2 + 0.015h + 3.04 - 2.97 = 0\)

- Simplify the Equation

Simplify the equation: \(-0.0006h^2 + 0.015h + 0.07 = 0\)

- Use the Quadratic Formula

Recall the quadratic formula: \(h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Identify \(a, b,\) and \(c\) from the equation \(-0.0006h^2 + 0.015h + 0.07 = 0\), where \(a = -0.0006\), \(b = 0.015\), and \(c = 0.07\)

- Calculate the Discriminant

Calculate the discriminant: \(b^2 - 4ac\):\(0.015^2 - 4(-0.0006)(0.07)\)\(= 0.000225 + 0.000168\)\(= 0.000393\)

- Apply the Quadratic Formula

Substitute the values into the quadratic formula: \(h = \frac{-0.015 \pm \sqrt{0.000393}}{2(-0.0006)}\)Simplify the expression: \(h = \frac{-0.015 \pm 0.01983}{-0.0012}\)

- Solve for h

Solve for both values of \(h\):\(h_1 = \frac{-0.015 + 0.01983}{-0.0012} \approx -4.025\) (ignore, not a valid solution)\(h_2 = \frac{-0.015 - 0.01983}{-0.0012} \approx 29.86\)

- Round to the Nearest Whole Hour

Round the valid solution, \(h_2 = 29.86\), to the nearest whole number: \(h \approx 30\) hours

Key concepts

GPA Calculation

The Grade Point Average (GPA) is a numerical representation of a student's academic performance. It is calculated by averaging the final marks obtained in various courses, usually on a scale of 0 to 4.0. For instance, in this exercise, we use a quadratic equation to estimate a student's GPA based on hours worked.
The given equation is: - \(g = -0.0006h^2 + 0.015h + 3.04\)
Here, the GPA (\(g\)) is modeled as a function of hours worked (\(h\)). You input the GPA value and solve for the number of hours worked to obtain an equation in standard quadratic form. Understanding how to plug in values and rearrange the equation is essential for accurate GPA calculation.

Student Workload

Student workload refers to the amount of time and effort a student devotes to academic and non-academic activities. Balancing these activities can impact both academic performance and personal well-being.
In the context of the given quadratic equation, the workload is represented by the variable \(h\) (hours worked each week). If a student works too many hours, it may negatively affect their GPA, as seen through the quadratic nature of the relationship. On the other hand, a moderate number of working hours might positively contribute to a higher GPA by helping the student manage responsibilities efficiently.
Therefore, understanding student workload helps in:

• Identifying optimal study and work hours balance for better GPA.
• Assessing the impact of extracurricular activities.
• Promoting overall student health and success.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that satisfy the equation. A quadratic equation typically has the form:
\(ax^2 + bx + c = 0\)
In the context of this problem, the equation becomes:
\(-0.0006h^2 + 0.015h + 0.07 = 0\)
To solve this, use the quadratic formula:\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Following the steps:

• Identify \(a, b,\) and \(c\) (here \(a = -0.0006, b = 0.015, and c = 0.07)\)
• Calculate the discriminant: \(b^2 - 4ac\)
• Substitute values into the formula and solve for \(h\)

For example, solving our equation, we found that:\(h_2 \approx 29.86\), which rounds to approximately 30 hours. This solution shows the importance of quadratic equations in real-life situations, such as estimating student workload for optimal GPA performance.