Question

The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $27 a bushel if she sells them now, with each tree yielding an average of 7 bushels. The yield increases by half a bushel per week for the next 5 weeks, but the price per bushel decreases by $1.50 per bushel each week. In how many weeks should the oranges be picked for maximum return?

Answer 1

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

=> $y = (27-1.5x)(7+0.5x)$

⇔$y= 27 (7 +0.5x) - 1.5x(7+0.5x) = 189 + 13.5x - 10.5x - 0.75x^{2}$

⇔$y = -0.75x^{2} +3x +189$

We have: if the equation has the form of $y =ax^{2} +bx +c$ with a≠0, its maximum value is: $max y = c - \frac{b^{2} }{4a}$

In the equation $y = -0.75x^{2} +3x +189$, we have: a = -0.75; b = 3; c = 189

=> $max y = c -\frac{b^{2} }{4a} = 189 - \frac{3^{2} }{4.(-0.75)} = 189 - \frac{9}{-3} = 189 - (-3) = 189+3 = 192$

To look for the number of weeks, we should find x (0≤x≤5) with which y = 192

$192 = -0.75x^{2} +3x +189$

⇔$-0.75x^{2} + 3x + 189 - 192 = 0$

⇔$-0.75 x^{2} + 3x - 3 =0$

⇔$-0.75x^{2} + 4*0.75x - 0.75*4 =0$

⇔$x^{2} -4x + 4 = 0$

⇔ $(x-2)^{2} = 0$

⇔ x = 2

The oranges should be picked in 2 weeks for maximum return