A horticulturist has just set up vegetable cultivation in a greenhouse where he grows peppers and tomatoes using sustainable energy sources, electricity for lighting and thermal energy for heating. He grows x units of peppers and y units of tomatoes. The entire amount that the farmer grows is sent to the market. The cost function of the crop is described as follows equation: ?(?, ?) = 12?2 + 24?2 The lighting's capacity means that it cannot grow more than 5,000 units in total per month. a. What should be the division between the production of peppers and tomatoes to minimize cost per month with the aim of always fully utilizing the capacity of the greenhouse? What is this minimum cost per month? b. Explain the significance of calculating the Lagrange multiplier i.e. interpret and at the same time discuss how it can be used to calculate if more productive lighting is obtained in the greenhouse and the cultivation will be 5,750 units per month.
A horticulturist has just set up vegetable cultivation in a greenhouse where he grows peppers and tomatoes using sustainable energy sources, electricity for lighting and thermal energy for heating. He grows x units of peppers and y units of tomatoes. The entire amount that the farmer grows is sent to the market. The cost function of the crop is described as follows equation:
?(?, ?) = 12?2 + 24?2
The lighting's capacity means that it cannot grow more than 5,000 units in total per month.
a. What should be the division between the production of peppers and tomatoes to minimize cost per month with the aim of always fully utilizing the capacity of the greenhouse? What is this minimum cost per month?
b. Explain the significance of calculating the Lagrange multiplier i.e. interpret and at the same time discuss how it can be used to calculate if more productive lighting is obtained in the greenhouse and the cultivation will be 5,750 units per month.
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