Hello, please help me.
There is a question where I have to optimize the area of a field that is being fence in like this: [|] i.e. two rectangle fields, side by side: area= 2x by y. I get to use 200ft of fencing. From that diagram and the amount of fencing I have, I made this equation: 200 = 4x +3y and A=2xy
from that area equation, I solved for A'(x) and got the critical number x=25 and noted that it is a (relative?) maximum.
From that, I solved for A subbing in x=25 and y=100/3 into A=2xy, where I got 1666.67 ft^2
Now this is where I stuck. I want to apply the absolute extrema test, but I'm not sure how to find out what the closed interval is. I'm pretty sure it has to do with the equation 200=4x +3y, but I have no idea how to go about it. Could you please show me how?
Again, thank you everyone who has helped me study for this test, I really appreciate it.
Since both x and y have to be positive numbers, you simply have to find for what values 200 = 4x+3y lies in the first quadrant.
Find the x and y intercepts
let x = 0 , y = 200/3
let y = 0 , x = 50
so 0 < x < 50
0 < y < 200/3
To find the closed interval for applying the absolute extrema test in this problem, you need to consider the constraints given, which are the equation for the perimeter of the field and the fact that both x and y must be greater than zero.
You mentioned that you already have the equation for the perimeter, which is 200 = 4x + 3y. To determine the closed interval, you need to find the range of valid values for x and y.
Let's consider the constraint that both x and y must be greater than zero. This means that the dimensions of the rectangular fields cannot be negative or zero.
From the equation 200 = 4x + 3y, we can rearrange it to solve for y in terms of x:
3y = 200 - 4x
y = (200 - 4x) / 3
Now, let's look at the constraints:
1. Both x and y must be greater than zero.
This means that (200 - 4x) / 3 must be greater than zero. To solve for x:
(200 - 4x) / 3 > 0
200 - 4x > 0
-4x > -200
x < 50
So, x must be less than 50.
2. The total length of fencing is 200 feet.
The perimeter equation gives us the relationship between x and y, and we can use it to find the values of y:
200 = 4x + 3y
3y = 200 - 4x
y = (200 - 4x) / 3
Now, the range of valid values for x and y can be determined by substituting x = 0 and x = 50 into the equation for y:
When x = 0:
y = (200 - 4(0)) / 3
y = 200 / 3
When x = 50:
y = (200 - 4(50)) / 3
y = 50 / 3
Therefore, the closed interval for x is (0, 50), and for y, it is (200/3, 50/3).
Now, you can apply the absolute extrema test within this closed interval to find the maximum or minimum values of the area.