SOLUTION: The owner of the local apple orchard estimates that if 24 trees are planted per acre, then each mature tree will yield 648 apples per year. for each additional tree planted per acr
Question 1198520: The owner of the local apple orchard estimates that if 24 trees are planted per acre, then each mature tree will yield 648 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 the most apples per year? Found 2 solutions by ikleyn, Theo:Answer by ikleyn(52781) (Show Source):
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The owner of the local apple orchard estimates that if 24 trees are planted per acre,
then each mature tree will yield 648 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 the most apples per year?
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If the number of trees is 24+n, then the number of apples is 648-12n in each tree.
Total number of apples is thus (24+n)*(648-12n). (1)
We want to find "n" such that the quadratic function (1) has maximum value.
The maximum of a quadratic form is reached half-way between its zeroes.
The zeroes of the function (1) are -24 and 648/12 = 54.
Half-way between these numbers is = = = 15.
The value of the quadratic function (1) is then (24+15)*(648-12*15) = 18252.
ANSWER. 24+15 = 39 trees and 18252 apples.
Solved.
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Be attentive when you interpret the obtained solution:
15 trees per acre should be added to existing 24 trees per acre,
making all planted trees number 39 per acre.
the total number of apples per year is equal to 24 * 648 = 15,552.
for each additional tree planted per acre, the number of apples produced by each tree decreased by 12 per year.
the formula for that is y = (24 + x) * (648 - 12x)
x is the number of additional trees per acre.
12x is the number of additional apples per tree per year.
y is the total number of apples per year.
when x is 0, the formula becomes y = 24 * 648 = 15552.
when x is 20, the formula becomes y = (24+20) * (648-240) = 44 * 408 = 17952
this is a quadratic equation.
simplify it to get:
y = 24*648 - 24*12*x + 648*x - 12*x^2
simplify to get:
y = 15552 - 288x + 648x - 12x^2
combine like terms and reorder in descending order of degree to get:
y = -12x^2 + 360x + 15552
when y = 0, this is in standard quadratic form of ax^2 + bx + c = 0
the equation will be maximum / minimum at x = -b/2a.
you get x = -360 / -24 = 360 / 24 = 15.
when x = 15, y = -12*15^2 + 360*15 + 15552 = 18252.
your solution is that the maximum number of trees per year will be obtained when 15 trees per acre are planted.
