Exercise 86Calculus › Single-variable calculus › Differentiation › Optimisation and curve sketchingHigh schoolAdded 4 Nov 2026

How many trees in the orchard?

#optimisation#modelling#quadratic function

Statement

An orchard has $50$ apple trees, each giving $800$ apples per year. Each additional tree planted reduces the yield of every tree by $10$ apples (they compete for light and water). How many trees should be added to maximise the total harvest?

Hint

With $x$ extra trees there are $50+x$ trees, each giving $800-10x$ apples.

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Solution

Total harvest. $H(x)=(50+x)(800-10x)=40\,000+300x-10x^2$ for $0\le x\le80$ (beyond $80$ the yield per tree would be negative).

Maximum. $H'(x)=300-20x=0\iff x=15$ . $H$ is a concave parabola, so this is the global maximum. Since $15$ is an integer, it is admissible as a number of trees.

Answer. Plant $15$ more trees: $65$ trees, each giving $650$ apples, for a total of $42\,250$ apples (compared with $40\,000$ at the start).

Remark. Whenever "more of something reduces the yield of each", the optimum balances the gain from the extra unit against the loss on all the others.

How many trees in the orchard?

Statement

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