Planting an apple orchard The owner of an apple orchard estimates that if $24$ trees are planted per acre, then each mature tree will yield $600$ apples per year. For each additional tree planted per acre, the number of apples produced by each tree decreases by $12$ per year. How many trees should be planted per acre to obtain at least $16, 416$ apples per year?

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Solution

Finding number of trees that should be planted per acre to obtain at least $16, 416$ apples per year:
Let $x$ be the number of extra trees planted to the existing $24$ trees
This means $x$ is the number of $12$ apple decreases from the original yield of $600$
Total number of apples will be $=$ $(24 + x) (600 - 12 x)$
We want at least $16, 416$ apples per year, therefore:
$(24 + x) (600 - 12 x) = 16416 14400 + 312 x - 12 x^{2} = 16416 \Rightarrow - 12 x^{2} + 312 x - 2016 = 0$
We know that,
if $a x^{2} + b x + c = 0$ be the quadratic equation then the quadratic formula be $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
By quadratic formula:
$a = - 12; b = 312; c = - 2016$
$x = \frac{- 312 \pm \sqrt{{(312)}^{2} - 4 \times (- 312) \times (- 2016)}}{- 24} = \frac{- 312 \pm \sqrt{97344 - 96768}}{- 24} = \frac{- 312 \pm \sqrt{576}}{- 24} = 13 \pm 1 = 12 or 14$
So, we have to plant $24 + 12 o r 24 + 14$ trees.
Hence, $36 o r 38$ trees should be planted per acre to obtain at least $16, 416$ apples per year.

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