Calculus I Applied Optimization Problem- How many trees should a farmer plant per acre to maximize her harvest?

Question

A farmer finds that if she plants 90 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest?

Answer 1

The total yield per acre as a function of trees planted per acre = the number of tree planted per acre times the number of bushels produced per acre.

y(t) = (90 + t)*(25 - 2t)

y(t) = 90*25 + 25t - 180t - 2*t*t

y(t) = 2,250 - 155t - 2t²

This is a quadratic function. The Precalculus non-derivative solution is t = -b/(2a). In this case, that is:

-(-155) / (-2*2) = -38.75

Since the coefficient of t² is negative, this indicates the function reaches a maximum at a negative x-value, and decreases from there.

If negative x-values are not allowed, we should pick 0 trees more, and plant 90 trees per acre. If we can decrease the number of trees planted, and the yield equation is still valid, then we should plant either 90 - 39 = 51 trees per acre or 90 - 38 = 52 trees per acre.

Note that we can't plant fractional trees; there must be a whole number of trees. So let us see what the two closest integers to -38.75 yield.

y(39) = (90-39)*(25-2(-39)) = 51*(25+2*39)

= 51* 103 = 5,253 bushels per acre.

y(38) = (90-38)*(25-2(-38)) = 52*(25+76)

= 52*101 = 5,252 bushels per acre.

In this case, plant 51 trees per acre to maximize the entire yield.

Compare this with 90 trees per acre: 90 * 25 = 2,250 bushels per acre. So, we get a much better yield with fewer trees planted. And undoubtedly, a lower cost with fewer trees planted per acre!

Now for the calculus solution: take the derivative and find its zero(s).

y'(x) = -155 - 4t = 0

4t = -155

t = -38.75, exactly the same as before.