Word problems : quadratic functions

[someguy37]

1. A farmer grows orange trees. He has 20 trees/hectare with 300 oranges/tree. When adding another tree to a hectare, the amount of oranges decreases by ten. What is the max yield of trees. (They want to know how many trees they should have in a hectare to maximize their orange production.)

The Attempt at a Solution

I think I figured out what to do, but I'm just missing one piece. So, you'll have 2 linear equations when multiplied together will give a quadratic function. For the amount of oranges per tree, I figured to be (300 - 10x). But as for the trees per hectare, I can't seem to get an equation. Very mind boggling.
Any help would be appreciated.
Thanks

 

[berkeman]

1. A farmer grows orange trees. He has 20 trees/hectare with 300 oranges/tree. When adding another tree to a hectare, the amount of oranges decreases by ten. What is the max yield of trees. (They want to know how many trees they should have in a hectare to maximize their orange production.)

The Attempt at a Solution

I think I figured out what to do, but I'm just missing one piece. So, you'll have 2 linear equations when multiplied together will give a quadratic function. For the amount of oranges per tree, I figured to be (300 - 10x). But as for the trees per hectare, I can't seem to get an equation. Very mind boggling.
Any help would be appreciated.
Thanks

Don't think your initial equation is quite right. Try filling in the numbers below to get a feel for the equations you want to write:

19*310=
20*300=
21*290=
etc.
25*250=
26*240=
27*230=
28*220=
etc.

 

[someguy37]

Ok.

5890
6000
6090
6160
6210
6240
6250
6240
6210
6160
...
I see the pattern, but I'm having a lot of difficulty putting it in an equation.

 

[berkeman]

Good, you at least see the pattern, and how there is a maximum number of oranges at 25 trees per hectare. You should also see kind of how to set up an equation for the number of oranges per hectare. Let T be the number of trees, and N be the number of oranges per hectare. Try writing the equation for N in terms of T, using the form that I posted the example calculations in... show us what you get.

BTW, to maximize the number N, what technique are you taught? Do you use differentiation, or do you have a different technique that you are learning now?

 

[someguy37]

Well, we're supposed to derive a quadratic equation from the problem. After having that, we use (-b)/2a to find the value of x, which gives the max.


As for the equation, I get

N = T * 300...but I know the 300 part is wrong. I'm missing something that isn't clicking.

 

[berkeman]

Well, we're supposed to derive a quadratic equation from the problem. After having that, we use (-b)/2a to find the value of x, which gives the max.


As for the equation, I get

N = T * 300...but I know the 300 part is wrong. I'm missing something that isn't clicking.

You're missing the variability with the number of trees T. You are told that for each tree over 20, you lose 10 oranges per tree. You should be able to write the resulting equation...

That equation does result in a quadratic in T...

 

[someguy37]

I'm sorry, but it's really just not clicking. This is Grade 11 Pre-Cal.

N = (20 + T)(300 - 10T)

 

[someguy37]

I'm not wanting you to give me the answer, honestly, but is there another way of giving me a hint?

 

[berkeman]

I'm sorry, but it's really just not clicking. This is Grade 11 Pre-Cal.

N = (20 + T)(300 - 10T)

Try using the equations that I posted, and talking through the general equation...

N = the number of trees T, multplied by 300 oranges per tree, less 10 oranges for each tree over 20...

N = T * (300 - ____ ( ______ ))

 

[someguy37]

ok, I got it now! Still, I find these sort of problems very hard to figure out. Are there steps I can follow to make it easier?

 

[someguy37]

Thank you very much! I really appreciate the help!

 

[berkeman]

ok, I got it now! Still, I find these sort of problems very hard to figure out. Are there steps I can follow to make it easier?

Good, glad you figured it out.

As for steps for setting up equations in word problems, one help is to work out a few examples of the numbers, like I showed in my first post above. That helps you start to get a feel for what is going on, and heck in this case, showed us both the answer right off the bat (so we could check our answer later).

Then, define the variable names, and sound out in words what you are trying to write as an equation. In this case, we wanted the number of oranges, and we knew it was based on the number of trees, and modified by another factor that also depended on the number of trees.

In more complicated word problems, you will end up with some number of equations and some number of unknowns (usually equal, if you have figured out all the applicable equations). One trick in those problems is to see how the problem is really "physical", that is, the equations you write constrain and define the problem well enough so that there is only one solution. Like when two lines intersect, if you constrain the equations of the two lines, you are physically constraining where they intersect. The intersection point cannot move anywhere, without changing part of the definition like an angle. Just like if you build a triangle out of boards, where the two far boards come together is defined by the lengths of the boards.

Hope that helps. Another thing that helps is to do as many word problems as you can. The more you do, the more comfortable and better you will get. Do all the problems in the chapter if you can, even if they're not homework. Or at least the last few hard ones, as practice.