Answer:
The oranges should be picked in 2 weeks for maximum return
Explanation:
We assume that the return of the owner is y ($)
Assume that the number of weeks the oranges should be picked to have maximum return is x (weeks). (x≥0)
If collect now, the price for each bushel is $27
As the price per bushel decrease by $1.50 per bushel each week
=> After x weeks, the price of a bushel decrease: 1.5x ($)
=> The price of 1 bushel after x weeks is: 27 - 1.5x ($)
If collect now, each tree can yield 7 bushels
As the yield increases by half a bushel per week for the next 5 weeks
=> After x weeks with x ≤ 5, each trees would yields: 7 + 0.5x (bushels)
The return = The price of each bushes × The quantity of bushels
=> [tex]y = (27-1.5x)(7+0.5x)[/tex]
⇔[tex]y= 27 (7 +0.5x) - 1.5x(7+0.5x) = 189 + 13.5x - 10.5x - 0.75x^{2}[/tex]
⇔[tex]y = -0.75x^{2} +3x +189[/tex]
We have: if the equation has the form of [tex]y =ax^{2} +bx +c[/tex] with a≠0, its maximum value is: [tex]max y = c - \frac{b^{2} }{4a}[/tex]
In the equation [tex]y = -0.75x^{2} +3x +189[/tex], we have: a = -0.75; b = 3; c = 189
=> [tex]max y = c -\frac{b^{2} }{4a} = 189 - \frac{3^{2} }{4.(-0.75)} = 189 - \frac{9}{-3} = 189 - (-3) = 189+3 = 192[/tex]
To look for the number of weeks, we should find x (0≤x≤5) with which y = 192
[tex]192 = -0.75x^{2} +3x +189[/tex]
⇔[tex]-0.75x^{2} + 3x + 189 - 192 = 0[/tex]
⇔[tex]-0.75 x^{2} + 3x - 3 =0[/tex]
⇔[tex]-0.75x^{2} + 4*0.75x - 0.75*4 =0[/tex]
⇔[tex]x^{2} -4x + 4 = 0[/tex]
⇔ [tex](x-2)^{2} = 0[/tex]
⇔ x = 2
The oranges should be picked in 2 weeks for maximum return
