Problem 27 - Algebra 2 and Trigonometry

Question

Based on data from a local college, a statistician determined that the average student's grade point average \((\mathrm{GPA})\) at this college is a function of the number of hours, \(h,\) he or she studies each weck. The grade point average can be estimated by the function \(\mathrm{G}(h)=0.006 h^{2}+0.02 h+1.2\) for \(0 \leq h \leq 20\) . a. To the nearest tenth, what is the average GPA of a student who does no studying? b. To the nearest tenth, what is the average GPA of a student who studies 12 hours per week? c. To the nearest tenth, how many hours of studying are required for the average student to achieve a 3.2 GPA?

Step-by-Step Solution

Answer

a. 1.2 b. 2.3 c. Approximately 16.7 hours

Step 1: Evaluate GPA for 0 Hours

To find the average GPA for a student who does no studying, substitute \( h = 0 \) into the function \( G(h) = 0.006h^2 + 0.02h + 1.2 \). This gives:\[ G(0) = 0.006(0)^2 + 0.02(0) + 1.2 = 1.2 \]Thus, the average GPA is 1.2.

Step 2: Calculate GPA for 12 Hours

Substitute \( h = 12 \) into the GPA function to find the average GPA for a student who studies 12 hours per week:\[ G(12) = 0.006(12)^2 + 0.02(12) + 1.2 \]Calculating, we have:\[ G(12) = 0.006(144) + 0.24 + 1.2 \]\[ G(12) = 0.864 + 0.24 + 1.2 \] \[ G(12) = 2.304 \]Rounded to the nearest tenth, the average GPA is 2.3.

Step 3: Solve for Hours Needed for 3.2 GPA

To find how many hours are needed to achieve a GPA of 3.2, set \( G(h) = 3.2 \) and solve for \( h \):\[ 0.006h^2 + 0.02h + 1.2 = 3.2 \]Subtract 1.2 from both sides: \[ 0.006h^2 + 0.02h - 2 = 0 \]Use the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 0.006 \), \( b = 0.02 \), and \( c = -2 \):\[ b^2 - 4ac = (0.02)^2 - 4(0.006)(-2) = 0.0004 + 0.048 = 0.0484 \]\[ h = \frac{-0.02 \pm \sqrt{0.0484}}{0.012} \] Calculating the square root and simplifying:\[ h = \frac{-0.02 \pm 0.220 \sqrt{}}{0.012} \] Positive solution :\[ h = \frac{0.22 - 0.02}{0.012} = \frac{0.20}{0.012} \approx 16.7 \]Approximately 16.7 hours of studying are required to achieve a GPA of 3.2.

Key Concepts

Understanding GPA Calculation

Grade Point Average, commonly known as GPA, is a factor in evaluating academic success. It's a numerical representation of a student's average performance. In this scenario, GPA is calculated based on study hours per week using a quadratic function. The given function is \[ G(h) = 0.006h^2 + 0.02h + 1.2 \] where:

• \( h \) is the number of hours studied per week.
• The terms involve constant coefficients indicating how each element affects the GPA.

When we plug in a specific value of \( h \), we can predict the average GPA. For example, without any study (\( h = 0 \)), the formula simplifies to an average GPA of 1.2. This initial value shows the base academic performance expected from students who don’t study. More study hours, like 12 hours per week, naturally increase this value according to the formula. This results in a more favorable GPA, 2.3, highlighting the positive correlation between study time and GPA improvement.

Exploring the Quadratic Formula

The quadratic formula is crucial for solving polynomial equations of degree two. It finds the variable when a quadratic equation is set to zero. Our primary equation turned into a quadratic form in step three, where we set \[ G(h) = 3.2 \] and derived \[ 0.006h^2 + 0.02h - 2 = 0 \].The quadratic formula: \[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] helps find the solutions for \( h \) where

• \( a = 0.006 \)
• \( b = 0.02 \)
• \( c = -2 \)

The discriminant (\( b^2 - 4ac \)) tells us the number of solutions and their nature. - A positive discriminant results in two real solutions.- A zero discriminant gives one real solution.- A negative discriminant indicates no real solutions.In this problem, the positive discriminant implies two possible solutions, reflecting variations in studying duration affecting GPA. For achieving a GPA of 3.2, we solve for \( h \) using the quadratic formula. Given two potential \( h \) values, we choose the one that fits our practical range of study hours.

Principles of Mathematical Modeling

Mathematical modeling involves representing real-world situations with mathematical equations. In this case, we used a quadratic function to model how study hours affect GPA, transforming an academic scenario into a mathematical equation. It provides a systematic way to predict outcomes based on assumed relationships. Here, gathering data on student performance provided a basis to construct the model. Allowing us to conclude the GPA pattern over a range of study hours - less than 0 is not feasible - exceeding 20 might not align with typical schedules. It gives educational planners a tool to provide targeted advice on study habits. By suggesting study hour increments, it shows improved performance while highlighting diminishing returns as study time increases significantly. This understanding aids in setting manageable goals and understanding performance expectations, making mathematical modeling influential in academic planning and personal study strategy development.