Question

Jacob currently grows and sells 10 butternuts each week to a spaza shop. Each butternut costs R4 to grow and is sold for R7. Jacob notices that the spaza shop does not yet sell cauliflowers. Cauliflowers cost R6 each to grow. Jacob visited the spaza shop and discussed the possibility of selling cauliflowers with the owner. He is open to the idea and is prepared to buy one cauliflower a week for R12,50. He could buy more cauliflowers, but is then prepared to pay less for each of the cauliflowers as he would not be able to sell them all at such a high price. After thinking about it, he is prepared to buy the following quantities of cauliflowers per week at the following prices: Prepared to buy 1 2 3 4 5 6 7 8 9 10 Price per cauliflower bought R12.50 R12 R11.50 R11 R10.50 R10 R9.50 R9 R8.50 R8 Jacob can only grow and sell 10 vegetables in a week. What is the optimum number of cauliflower and butternut he should sell to maximize his profit?

Answer 1

Answer:

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Answer 2

given profit per butternut and table of prices for different number of cauliflowers per week, find optimal number of butternuts and cauliflowers to get maximum profit

Explanation:

1. let the number of butternuts and cauliflower grown per week be 'b' and 'c' respectively. then we have , $b+c=10$   ---(i)
2. given that cost growing and selling a butternut is $4\ and\ 7$ respectively. hence, the profit per butternut is $3$.
3. now let the profit per cauliflower be 'p' then total profit in a week is given by,  $T=3b+pc$                                                                                                                     from (i) we get, $T=3(10-c)+pc$     -> $T=30+(p-3)c$                                          
4. now for maximum profit we need ,                                                                           $(30+(p-3)c)_{max}\\((p-3)c)_{max}$                ---(i)
5. now given cost of growing a cauliflower $6$ and different prices per cauliflowers we have,                                                                                                     $c-\ \ \ \ \ \ 1\ \ \ \ \ \ \ \ 2\ \ \ \ 3\ \ \ \ \ \ \ 4\ \ \ \ \ \ 5\ \ \ \ \ \ 6\ \ \ \ 7\ \ \ \ \ 8\ \ \ \ 9\ \ \ \ \ \ 10\\price-\ 12.5\ \ \ 12\ \ \ 11.5\ \ \ 11\ \ \ \ 10.5\ \ \ 10\ \ \ 9.5\ \ \ 9\ \ \ 8.5\ \ \ \ 8\\p- \ \ \ \ \ \ 6.5\ \ \ \ \ 6\ \ \ \ \ 5.5\ \ \ \ \ 5\ \ \ \ 4.5\ \ \ \ \ 4\ \ \ \ 3.5\ \ \ 3\ \ \ 2.5\ \ \ \ 2\\c(p-3)\ 3.5\ \ \ \ \ 6\ \ \ \ \ 7.5\ \ \ \ 8 \ \ \ \ \ 7.5\ \ \ \ \ 6 \ \ \ \ 3.5\ \ \ 0\ -4.5\ -10$      ---(ii)
6. hence from (i) and (ii) we get  $((p-3)c)_{max} = 8$  for $c=4$                      
7. hence, for maximum profit Jacob should grow and sell $4$ cauliflowers and $6$ butternuts.