A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. she estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. how many trees should she plant per acre to maximize her harvest?

Question

Mathematics

Puddles

31 March, 01:04

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. she estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. how many trees should she plant per acre to maximize her harvest?

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Answers (1)

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Doll · Answer 1

= 50 trees

Step-by-step explanation:

Let F be the number of fruit per acre

F = (75 + x) * (20 - 3x)

F = 75 * 20 - 75 * 3x + 75 * x - 3x^2

F = 1500 - 225 x + 75x - 3x^2

F = 1500 - 150x - 3x^2

Factor out the 3 and complete the square (because this is a quadratic equation, we're going to find the vertex of the parabola):

F = - 3x^2 - 150x + 1500

F = - 3 * (x^2 + 50x - 500)

F = - 3 * (x^2 + 2 * 25x + 25^2 - 25^2 - 500)

F = - 3 * ((x + 25) ^2 - 500 - 625)

F = - 3 * ((x + 25) ^2 - 625)

F = - 3 * (x + 25) ^2 + 3375

So the vertex is at (-25, 3375)

75 - 25 = 50

The farmer should plant 50 trees per acre to maximize her yield