Question

PLEASE HELP!!!

Answer 1

A city wants to plant maple and spruce trees to absorb carbon dioxide. It has $2,100 to spend on planting spruce and maple trees. The city has 45,000ft^2 available for planting.

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 the city plant to maximize carbon dioxide absorption.

Answer 2

Your problem statement mentions "data from the table," but you have not posted such a table. Could you do that now?

Answer 3

Yes, just a second.

Answer 4

Spruce Maple
Planting Cost $30 $40
Area Required 600ft^2 900ft^2
Carbon Dioxide Absorption 650lb/year 300lb/year

Answer 5

@mathmale

Answer 6

The problem statement doesn't say so, but you really need to choose variables to represent the number of spruce trees and the number of maple trees. Once you have these variables in mind, you can write inequalities and equations involving them. Which letters are y ou choosing to represent these two different counts?

Answer 7

Let X = Spruce
Let Y = Maple

Answer 8

Or, use s for # of spruce trees and m for # of maple trees.

Answer 9

I want to represent with x and y.

Answer 10

It's easier for me.

Answer 11

Fine, be my guest, Braydon.

Answer 12

What does "objective function" mean to you, and what is that function in this case? What are you measuring?

Answer 13

What's the objective of planting trees?

Answer 14

\[30x + 40y \le 2100\]
\[600ft^2x + 900ft^2 \le 45000ft^2\]
\[X \ge 0\]
\[Y \ge 0\]

Answer 15

Sorry, that was A.

Answer 16

Good...you do know a lot already.
BUT...why go thru all this trouble? What's the objective here? You haven't mentioned it.

Answer 17

B. P = 650x + 300y

Answer 18

You're trying to minimize / maximize something. Which is it, and what is that something?

Answer 19

Maximize the amount of trees to plant for carbon dioxide absorption?

Answer 20

Not quite. Maximize ... what? Lose some of the words in your statement.

Answer 21

Maximize carbon dioxide absorption?

Answer 22

Exactly. Now we're right on target.

Answer 23

How would I graph the feasible statement?

Answer 24

You have typed in a number of constraints. Have you used all of the specifications given you?

Answer 25

...typed in a number of inequalities based upon the given constraints.

Answer 26

?

Answer 27

Let's let that go.

You have four inequalities. Next step is to graph them on a single set of coordinate axes. Could you do that using the Draw utility, below?

Answer 28

But, I have to solve them first. Don't I? For y = mx+b ?

Answer 29

You have to solve for the coordinates of their intersections, yes. Draw the constraint lines first and find those coordinates next, OR do that in reverse order. Either is fine.

Answer 30

I got these equations:
\[y \le \frac{ -3 }{ 4 } + \frac{ 105 }{ 2 }\]
\[y \le -\frac{ 2 }{ 3 } + 50\]

Answer 31

I'll take your word for them.
You'll need them.
BUT: My request was that you find the intersections of the four lines representing the constraints.

Answer 32

for example ...\[y \le -\frac{ 2 }{ 3 }x + 50~and~x \ge 0\] intersect at some point. What is that point? (0, ? )

Answer 33

Note that you'd left out the 'x' in the first inequality; I've put it in for you.

You have four lines. Again, your task is to determine the coord. of intersection. Ignoring (0,0), there will be four such points of intersection.

Answer 34

I can't graph this...

Answer 35

Why not?

Answer 36

I'm having troubles.

Answer 37

Using the example I gave you just above, we get the vertex (0, 50).

As for having troubles, you'll need to be more specific. What do you need help with?

Answer 38

Do you understand where I got that (0,50)?

Answer 39

Yes.

Answer 40

There's another vertex that's easy to find; it's on the x-axis. It looks like ( ? , ? ). Can you complete this work?

Answer 41

No.

Answer 42

Note: You've also left out the "x" here:\[y \le \frac{ -3 }{ 4 } + \frac{ 105 }{ 2 }\]

Answer 43

If you want the x-intercept, set y=0 and solve for x.

Take your own inequality, insert the x, set y = 0 and solve for x. Try it. Hint: LCD is 4.

Answer 44

Yes, I see that I left that out on both equations.

Answer 45

There's my graph. How would I move down 2 and right 3?

Answer 46

Nice that you can share your graph.
Assuming that your scale divisions all represent 1 unit, move down 2 squares and right 3 squares. I'm not doing the actual work, just interpreting what you are doing.

Answer 47

Could I move down the way it is or would I have to do something with the graph?

Answer 48

Stop for a minute: think. What is your goal at this point? Why are you finding these points of intersection?

Answer 49

To find a maximum.

Answer 50

You are to find four corner points, and then substit. the x-and y-coords. of each into the objective equation. The "winner" is the point whose coordinates present us with the largest pollutant absorption.

Answer 51

One such corner point is (0,50).

Answer 52

So, again, you need to find the coord. of all of the intersections of your four linear graphs.

Answer 53

How could I graph 52.5?

Answer 54

You need the value of the coordinate more than you need to worry about graphing it.

Answer 55

But if it's really 52.5, then you'd have to extend the x- and y-axes quite a bit, so that you could mark 52.5 on whichever axis applies.

Answer 56

Look at my graph. How could I graph 52.5??

Answer 57

If you start with \[30x + 40y \le 2100\]

Answer 58

I already graphed that.

Answer 59

and want to find the y intercept, you let x=0. Result: 40y=2100, or y=52.5.
That particular corner point is (0,52.5).

Answer 60

Wait, no I didn't.

Answer 61

You're not understanding my question?!

Answer 62

If you now want the x-intercept, let y=0:\[30x + 40y \le 2100\]

Answer 63

x is then 70. So another corner point is (70,0).

Answer 64

Slow down.

Answer 65

Where have I lost you?

Answer 66

I DON'T KNOW WHERE TO GRAPH 52.5!

Answer 67

On my graph, I am going up by 5's.

Answer 68

that 52.5 shows up in the corner point (0,52.5). That means you need to plot this point on the y-axis. Fine... if you let each square represent 5 units, then you have to go up 10 1/2 units to get to "52.5."

Answer 69

Let me redo my graph.

Answer 70

How can I set my graph up?

Answer 71

It's worth the time to do that, but I won't necessarily be able to wait until you've finished that.

Once again, you have to find a bunch of corner points. I believe there are four, including (0,0), (0,52.5), (70,0) and the intersection of the two inequality equations.

If you can solve simultaneously the 2 inequality equations involveing x and y both, you will have found the fourth corner point.

Answer 72

(0,0)
(0,50)
(30,30)
(70,0)

Answer 73

You need only one more corner point: the intersection (solution ) of

Answer 74

\[30x + 40y \le 2100\] and \[600ft^2x + 900ft^2 \le 45000ft^2\]

Answer 75

I 've solved the above system; the y-coordinate of the solution is 30. You find the x-coordinate.

Having done that you will now have 4 corner points to test in the objective function.

Answer 76

650x + 300y is your objective function.

One by one, take your four corner points, or more specifically, take the coordinates of each one and subst. them into this objective function. Which set of coordinates results in the largest value of the ojb. function?