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A farmer owns $50$ papaya trees. Each tree produces $600$ papayas in a year. For each additional tree planted in the orchard, the output of each tree (including the pre-existing ones) drops by $5$ papayas. How many trees should be added to the existing orchard in order to maximize the total production of papayas?
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Answer -  35 

 

Form a function from the above data in x . Find it's maxima.

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Let us add $p$ additional papaya trees.

Total number of papayas will be given by,

$f(p) = 50\times 600 + p\times 600-(50+p)\times 5\times p \\ \qquad = 30000 + (600-250)p - 5p^2 \\ \qquad = 30000 + 350p - 5p^2$

To find maxima, differentiate $f(p)$ w.r.t  $p$.

${f}'(p) = 350 – 10p = 0 \\ \quad \implies 10p = 350 \implies p = 35$

Therefore, we need to add 35 extra trees to maximize the total production of papayas.

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