Alright, so I recently started my new semester, and I've forgotten almost everything that I've learned the previous year regarding math. I was assigned to do various questions, and I'm not exactly sure how to do it. The question is:
Rebecca has 600 m of fencing for her cornfield. The creek that goes through her farmland will form one side of the rectangular boundary. Rebecca considers different widths to maximize the area enclosed.
A. What are the minimum and maximum values of the width of the field?
I'm assuming that I will need to create a parabola for this question, due to the question asking a minimum and maximum. I also have the feeling completing the square will be involved, but I pretty much forgot how to do both. But here's what I got so far:
P = L + 2W (the creek makes up 1 length?)
L = 600 - 2W
A = W X L
That's all I got for now... any help, please?
B. What equations describe each?
i) The relationship between the length and width of the field.
ii) The relationship between the area and width of the field.
C. Copy and complete these table of values for widths that go from the least to greatest possible values in intervals of 50 m.
Width (m) | Length (m) | Area (m^2)
P = L + 2W
L = 600 - 2W
A = W X L
A = W X (600 - 2W)
A = 600W - 2W^2
A = -2W^2 + 600W
A = -2(W^2 - 300W) -> (300/2)^2 = 22500
A = -2(W^2 - 300W + 22500 - 22500)
Alright, got this so far. But I'm not sure what to do from here. From what I remember, I will have to use the -2 again on the next line, and isolate the -22500, but how do I do this... if that's what I'm even supposed to do?
You really don't need to complete the square from here, it would just be making life difficult and who has time for that?
To solve for make
Take common factors from the RHS
Using the null factor law
What is half way between 0 and 300?
Put this value into for to get max area.
P = L + 2W
L = 600 - 2W
A = W X L
A = W X (600 - 2W)
A = 600W - 2W^2
0 = 600W - 2W^2
0 = 2W(300 - W)
Wait, where did the 0,300 come from? In order for the right side of the equation to = 0, W would have to = 300.
Half way between 0 and 300 = 150. Therefore W (min value?) = 150.
A = 600W - 2W^2
A = 600(150) - 2(150)^2
A = 90000 - 2(22500)
A = 90000 - 45000
A = 45000
Therefore max area is 45000. Though, the question is asking for the minimum and maximum of the width of the field. Is the max area the max value for the width?
Alright, thanks very much for the help.
Before even attempting to complete the square, I guessed that the minimum width was 1, and the maximum width was 299. I guessed 1 because the length could be 598, and the two widths would be one each, equaling 600 (perimeter), but I guess that was wrong.
The next question asks:
B. What equations describe each?
i) The relationship between the length and width of the field.
A = W X L
ii) The relationship between the area and width of the field.
W = A/L
These look correct?
Got it now, ty.
Question C asks:
C. Copy and complete these table of values for widths that go from the least to greatest possible values in intervals of 50 m.
Width (m) | Length (m) | Area (m^2)
Width (m)
0
50
100
150
200
250
300
Length (m)
600
500
400
300
200
100
0
Area (m^2)
0
25000
40000
45000
40000
25000
0
A friend shared these results with me. I don't even see any pattern within them at all. Are they correct? Will I have to somehow apply my results in questions A and B to complete this chart?