Question

A farmer finds that if she plants 70 trees per acre, each tree will yield 30 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

Answer 1

The farmer should plant 40 trees per acre to maximize her yield to harvest.

What is a system of equations?

A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.

Let F be the number of fruit per acre  

$F = (70 + x) \times (30 - 3x) \\F = 70 \times 30 - 70 \times 3x + 30 \times x - 3x^2 \\F = 2100 - 210 x + 30x - 3x^2 \\F = 2100 - 180x - 3x^2$

Factor out the 3 and complete the square

$F = -3x^2 - 180x + 2100\\F = -3 \times (x^2 + 60x - 700) \\F = -3 \times (x^2 + 2 * 30x + 30^2 - 30^2 - 700) \\F = -3 \times ((x + 30)^2 - 500 - 900) \\F = -3 \times ((x + 30)^2 - 400) \\F = -3 \times (x + 30)^2 + 1200$

So the vertex is at (-30, 1200)  

70 - 30 = 40  

The farmer should plant 40 trees per acre to maximize her yield to harvest.

Learn more about equations here;

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