The annual yield per walnut tree is fairly constant at 60 pounds per tree when the number of trees per acre is 20 or fewer. For each additional tree over 20, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding. Therefore, assuming that the number of walnut trees per acre is greater than 20,the annual yield of walnuts on the acre, A, as a function of the number of walnut trees, t, is the following:
A(t) = −2t^2 + 100t
(a) How many walnut trees should be planted per acre to maximize the annual yield for the acre?
(b) What is is the maximum number of pounds of walnuts per acre?
Recall that for the parabola y = ax^23+bx+c,
(a) The vertex is at x = -b/2a
(b) evaluate y at that value of x.
Or, think of the vertex form for this parabola
A(t) = -2(t-25)^2 + 1250
The vertex is at (25,1250)
To find the number of walnut trees that should be planted per acre to maximize the annual yield (A(t)), we need to find the maximum point of the given quadratic function.
(a) To find the number of walnut trees, t, that maximizes the annual yield, we need to identify the vertex of the quadratic function A(t) = -2t^2 + 100t.
The vertex of a quadratic function in the form A(t) = at^2 + bt + c is given by the formula:
t = -b / (2a)
In this case, a = -2 and b = 100, so substituting these values into the formula, we get:
t = -100 / (2*(-2))
t = -100 / (-4)
t = 25
Therefore, to maximize the annual yield for the acre, we should plant 25 walnut trees per acre.
(b) To find the maximum number of pounds of walnuts per acre, we need to substitute the value of t = 25 into the quadratic function A(t):
A(25) = -2(25)^2 + 100(25)
A(25) = -2(625) + 2500
A(25) = -1250 + 2500
A(25) = 1250
Hence, the maximum number of pounds of walnuts per acre is 1250 pounds.