Final answer:
To maximize the GPA with 15 hours available, an optimization problem needs to be solved using the function G = 1/8(2√e + √s), where e and s are study hours for economics and sociology. Calculus optimization is employed to express the GPA function in terms of either e or s, then derivatives are used to find the critical points which determine the optimal allocation of study time between the two subjects.
Step-by-step explanation:
The student's question revolves around an optimization problem where the goal is to maximize the GPA based on the given function G = 1/8(2√e + √s), with e and s representing the hours per day spent studying economics and sociology respectively. Given that the student has 15 hours available for study each day, we need to determine how many hours to allocate to studying each subject in order to achieve the highest possible GPA.
Given a fixed amount of study time (15 hours), we can set up the optimization problem as follows:
- Let e + s = 15 (since the total time for studying is 15 hours)
- Our objective is to maximize G = 1/8(2√e + √s)
To solve for the optimal allocation of study time, we use the method of calculus optimization. We can rewrite the constraint as s = 15 - e and substitute it into the GPA function to express it in terms of e alone:
G(e) = 1/8(2√e + √(15 - e))
Then we find the derivative of G with respect to e and set it to zero to find the critical points:
G'(e) = 1/8(1/√e - 1/2√(15 - e))
Setting G'(e) = 0 and solving for e will give us the number of hours to allocate to studying economics, and the remainder will be the hours allocated to sociology. Once we find e, we can compute s via the constraint. The second derivative test can be done to ensure the solution corresponds to a maximum.
The solution to this problem would provide the optimal study time allocation for economics and sociology in order to maximize the GPA.