Question

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?

Answer 1

Answer:

= 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

Answer 2

$50$ trees she should plant per acre to maximize her harvest.

$N=(75+x)\times (20-3x)$

where $N=$Number of fruits per acre

Solve the value of $x$

$N=75\times20-75\times3x+75\times x-3x^2$

$N=1500-225x+74x-3x^2$

$N=1500-150x-3x^2$

Rearrange the above equation,

$N=-3x^2-150x+1500$

$N=-3(x^2+50x-500)$

solve the above quadratic:

$N=-3(x+25)^2+3375$

so the vertex is at $(-25,3375)$

$75-25=50$

Hence ,$50$ trees she should plant per acre to maximize her harvest.

Learn more about quadratic equation here;

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