The number of apples produced per acre of land depends on how densely the trees are planted. if
n
trees are planted in an acre of land, the function that can be used to estimate the total number of apples that will be produced is the quadratic function ()
a
(
n
)
. one farmer found that he would get a maximum yield of 202,500 apples when he planted 450 apple trees per acre. when he increased the trees per acre to 900, the trees produced no apples.

A(n)=an^2+b^+c

a=-1

b=900

c=0

Step-by-step explanation:

We assume the function A(n) like this: A(n)=an^2+bn+c and we have two conditions to solve the problem:

1. If the farmer plant 450 trees for acre, the farmer obtain 202,500 apples, which is, in terms of the function:

If n=450, then A(n=450)=202,500   (Condition 1)

2. If the farmer plant 900 trees for acre, the farmer obtain 0 apples, which is, in terms of the function:

If n=900, then A(n=900)=0   (Condition 2)

But, although the problem does not talk about it, we have a third condition, because if the farmer does not plant any tree per acre, then, he will produce 0 apples, which is

If n=0, then A(n=0)=0   (Condition 3)

Using those conditions, we can start to find the value of the coefficients for the function. First, we are going to start for the simplest condition, which is condition 3. If n=0 then A(n) is:

A(n=0)=0

Then, for the definition of A(n),

a(0^2)+b(0)+c=0

So, we obtain that c=0, using the condition 3.

Now, we are going to use the condition 2 to obtain the value of the coefficient b. So, with n=900, A(n=900)=0

A(n=900)=a(900^2)+b(900)=0

b(900)=-a(900^2)

b=-900a   (Eq 1)

Then, we are going to use the condition 1 and the value for b obtained for the condition 2 (Eq 1). So, with n=450, A(n=450)=202,500, which is

A(n=450)=a(450^2)+b(450)=202,500

a(450^2)+(-900a)(450)=202,500

(450)(a(450)-900a)=202,500

450a-900a=202,500/450

-450a=450

a=-1   (Eq 2)

With the obtained value for the coefficient a, we clear the value of the coefficient b using the equation 1. So,

b=-900a

b=-900(-1)

b=900

And, those are the values for the coefficients of the A(n) function. And, the function is:

A(n) = -n^2+900n