Question

I've been stuck on this for a long time and wonder if someone could help walk me through it:
A city wants to plant maple and spruce trees to absorb carbon dioxide. It has $2100 to spend on planting trees. The city has 45,000ft2 available for planting.

Planting Cost:Spruce=$30 Maple=$40
Area Required:Spruce=600ft2 Maple=900ft2
Co2 Absorption:Spruce=650lb/yr Maple=300lb/yr
a.Use the data from the table. Write the constraints for the situation.
b.Write the objective function.
c.Graph the feasible region and find the vertices.
d.How many of each tree should be planted to maximize co2 absorption?

Answer 1

What part are you having trouble with? How to get started? A particular equation? How to interpret the results?

Answer 2

How to get started and also how to graph it. But I am just generally confused with this concept!

Answer 3

Is your data correct? I see no reason to plant any maple trees (higher cost, larger footprint, and lower carbon dioxide absorption).

Answer 4

Sorry for the long delay, but geoffb is correct. According to the constraints, there's no reason to plant any maple trees.
Can you go over your data?

Answer 5

Yes my data is correct. I agree that that would make the most sense but I still have to prove that by answering the questions, which I am confused on.

Answer 6

Say we call the number of maple trees x
Say we call the number of spruce trees y
Given that you know the absorption of co2 per tree you can setup an equation as:
300x+650y=CO2 absorption
The total absorption is the absorption per tree for each type of tree, times the number of trees planted of that type...

Answer 7

Next, set up the constraints equations:
COST
40x+30y<=2100
Again: cost per tree times number of trees of one type of tree PLUS cost per tree times number of trees planted of the second type.
AREA
900x+600y<=45000

Answer 8

Also, note an additional constraint:
x=>0 , y=>0
Next plot all 4 constraint equations:
You will end up with a plot of two down sloping lines, the intersection of the areas under these curves, (but above the x>0 and y>0) is the solution area.

Answer 9

To get the extreme where the lines cross, set them equal (if you have them as y=mx+b) and solve for x. This will give you the x value where they cross and then substitute and solve for y.

Answer 10

Finally, substitute these extremes into the equation you want to maximize and see which values give you the highest value. This means these extremes are the number of trees of each type you should plant.

Answer 11

The image shows the two lines, the area below BOTH of them, is the solution area.
The extremes give you the x,y values to sub into the equation you want to maximize.
When you sub them all, you find the highest value at the (30,30) point.
This means the optimal distribution is 30 maple and 30 spruce. For a highest value of 46,500 lb/yr of CO2

Answer 12

@misty497 Let me know if there's something you can't follow or would like further clarification.

Answer 13

butthole

Answer 14

Thanks bigrobot I was wondering what your nickname was!

Answer 15

Rudy, thank you so much for helping! For the equations, once you turn them into y=mx+b, would they be y<=-4/3x+70 and y<=-9/6x+75 or am I completely off??

Answer 16

Those are the same equations I got.

Answer 17

Okay thank you. I am also a little confused on what you mean when you were talking about the extremes?

Answer 18

The area under both lines along with the x and y axis form a square-ish figure.
According to this method, the maximums happen at the extremes of these figures. That's why we need to find the x and y coordinates of these corners. I listed those in the drawing. Those are the values you need to plug into the equation that you want to maximize (the one for co2 absorption) and see which one gives the maximum value for co2 absorption. The x,y pair that gave you the maximum value is the x and y amount of trees you need to plant.

Answer 19

Thank you so much for your help!!! I really understand it now.

Answer 20

Happy to help!