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Can You Factorize The Function?

A Florida citrus grower estimates that if 70 orange trees are planted, the average yield per tree will be 700 oranges. The average yield will decrease by 7 oranges per tree for each additional tree planted on the same acreage. Express the grower’s total yield as a function, y, of the number of additional trees planted, estimate the total number of trees the grower should plant to maximize yield.

asked 9 months ago

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taimur

Suppose y(t) is function.

Y(70) = 700
y(t) = y0 - 7(t -1), (where y0 is the yield of 1st tree)
substitute (70,700) gives
700 = y0 - 7(70-1)
700 = y0 - 490 + 7
y0 = 700 + 483
y0 = 1183
so
y(t) = 1183 - 7(t-1)
or y(t) = 1183 - 7T (where T = (t-1) each additional tree)

but y(t) is the average per tree so the total yield (y) would be
y = t (1183-7T)
since T = t-1
y = (T+1)(1183-7T)
y = -7T^2 + 1176T + 1183

To get maximum yield of trees, we require where the drivative = 0

now
y = t(1183-7(t-1))
y = t(1183 - 7t + 7)
y = -7t^2 + 1190t
differentiate
dy/dt = -14t + 1190
dy/dt = 0 gives
-14t + 1190 = 0
14t = 1190
t = 1190/14 = 85

ANSWER :::: 85 trees will give maximum yeild

answered 9 months ago

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