- #1
Knore88
- 14
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A student wishes to minimize the time required to gain a given expected average
grade, đť‘š, in her end-of-semester examinations. Let \(\displaystyle {t}_{i}\) be the time spent studying
subject i\(\displaystyle \in\){1,2}.
Suppose that the expected grade functions are \(\displaystyle {g}_{1}\)(\(\displaystyle {t}_{1}\)) = 40+8\(\displaystyle \sqrt{{t}_{i}}\) and \(\displaystyle {g}_{2}\)(\(\displaystyle {t}_{2}\)) = 2\(\displaystyle {t}_{2}\).
Thus the individual’s optimization problems is to choose \(\displaystyle {t}_{1}\) and \(\displaystyle {t}_{2}\) to minimize total studying time 𝑇 = \(\displaystyle {t}_{1}\) + \(\displaystyle {t}_{2}\) subject to obtaining a mean grade of 𝑚 where
đť‘š - \(\displaystyle \frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\) = 0
I need to write down the Lagrangian for the individual’s optimization problem and solve for the optimal choices of \(\displaystyle {t}_{1}\), \(\displaystyle {t}_{2}\) and 𝑇 in the case where the student wishes to obtain an expected mean grade of 70.
To be completely honest, I'm not sure where to start.. or really how to do the problem![Confused :confused: :confused:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
grade, đť‘š, in her end-of-semester examinations. Let \(\displaystyle {t}_{i}\) be the time spent studying
subject i\(\displaystyle \in\){1,2}.
Suppose that the expected grade functions are \(\displaystyle {g}_{1}\)(\(\displaystyle {t}_{1}\)) = 40+8\(\displaystyle \sqrt{{t}_{i}}\) and \(\displaystyle {g}_{2}\)(\(\displaystyle {t}_{2}\)) = 2\(\displaystyle {t}_{2}\).
Thus the individual’s optimization problems is to choose \(\displaystyle {t}_{1}\) and \(\displaystyle {t}_{2}\) to minimize total studying time 𝑇 = \(\displaystyle {t}_{1}\) + \(\displaystyle {t}_{2}\) subject to obtaining a mean grade of 𝑚 where
đť‘š - \(\displaystyle \frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\) = 0
I need to write down the Lagrangian for the individual’s optimization problem and solve for the optimal choices of \(\displaystyle {t}_{1}\), \(\displaystyle {t}_{2}\) and 𝑇 in the case where the student wishes to obtain an expected mean grade of 70.
To be completely honest, I'm not sure where to start.. or really how to do the problem