# Thread: Word problem finding an equation

1. ## Word problem finding an equation

Not too sure if this problem belongs in this section of the forum.

A lemon tree will produce 300 lemons per year if 16 trees are planted. For every additional tree planted, the yield per tree decreases by 10 lemons per year.

If an additional x trees are planted, show that the number of lemons produced will be:

$N=-10x^2+140x+4800$

2. What have you tried to do so far or are you completely lost?

3. I'm really lost:

I know that if 16 trees are planted, that there are 300 x 16 = 4800 lemons

But what confuses me is how they get -10x^2+140x

I would have thought that you could just do, N = 4800 - 10(16+x)

4. You are NOT completely lost. You've got the first part right. There are indeed 4800 lemons produced when you have 16 trees. This corresponds to x = 0.

Your mistake is in the second part. You cannot just get $N = 4800 - 10(16+x)$ because once you plant a tree, the yield of EVERY OTHER tree decreases by 10 lemons. If x = 1, that means that you have 17 trees, each at 290 lemons per year. Once you have x = 2, you have 18 trees at 280 lemons per year. Lets see if we can make an equation out of this.

Lets designate the number of lemons a tree makes is $T(x) = 300 - 10x$. That is when we plant an additional x trees, the rate of lemon production of each tree reduces by 10 lemons per year. Do you agree?

Now that we have the rate of lemons per year, how many trees do we have? We started with 16 and we got an additional x trees. This gives us 16+x trees. Now the number of lemons produced each year is simply (16+x)T(X), which is just the number of trees multiplied by the number of lemons each tree produces. Lets multiply them out!

$\displaystyle (16+x) T(x) = (16+x)(300 - 10x) = 16.300 + 300.x - 16.10x-10x.x = 4800 +300x - 160x-10x^2 = -10x^2+140x+4800$

Is this clear? If you have any questions, ask!

5. EDIT: Double post. I don't know how this happened!