How would I find total yield for an acre in this question?

Question

“The annual yield per pear tree is fairly consistent at 250 pounds per tree when the number of trees per acre is 30 or fewer. For each additional tree over 30, the annual yield per tree for all trees on the acre decreases by 15 pounds due to overcrowding.

a. Express the yield per tree, Y, in pounds, as a function of the number of pear trees per acre, x.

b. Express the total yield for an acre, T, in pounds as a function of pear trees per acre, x.”

Question is word for word. Thanks in advance!

Answer 1

Due to the nature of the problem (the trees have a fixed yield to a certain point, after which the yield changes) it seems like the answer should be expressed as a piecewise function.

So we have three variables here: x (the number of trees per acre), Y (the yield per tree in pounds), and T (the total yield per acre).

In Part A we are asked to find Y, the yield per tree in pounds. This is where the piecewise function first comes in. Since the yield per tree is stable up to 30 trees, we would write it like this: Y = { 250 , x ≤ 30 (meaning, when x is less than or equal to 30, Y is equal to 250).

Since the yield changes for every tree greater than 30, we have to find the equation. So you start with 250 (Y={250), then you subtract 15 (Y={250-15) for every tree (Y={250-15(x)) greater than 30 (Y{250-15(x-30)).

Now let's check that. Since x will be a number greater than 30 (eg. 35), we can plug that into our equation to get 250-15(35-30), which simplifies to 250-15(5), simplifies again to 250-75, which equals to 175.

That worked, so the finished piecewise function for Part A would be:

Y = { 250 , x ≤ 30

{ 250 - 15(x - 30) , x > 30

So that's Part A done. Part B is now a lot simpler. Part B wants the total yield per acre, which is going to be equal to the yield of each tree multiplied by the number of trees per acre. Therefore, T can be written as:

T = { 250x , x ≤ 30

{ [250 - 15(x - 30)]x , x > 30

or

T = { 250x , x ≤ 30

{ 250x - 15x² - 30x , x > 30

or

T = Yx

I hope that helps!