a total of 8 meters of fencing are going to be used to fence in a rectangular cage for pets and divide it into three smaller cages
**the photo that is shown with this question is three (3) boxes stuck together side by side horizontally***
determine the overall dimensions that will yield the maximum total enclosed area.
To determine the overall dimensions that will yield the maximum total enclosed area, we can use the concept of calculus optimization.
Let's assume that the length of the rectangular cage is "L" and the width is "W". Since there are three smaller cages, each cage would have a width of W/3.
Now, let's calculate the perimeter of the rectangular cage using the given information:
Perimeter = 2(L + W)
Since we have a total of 8 meters of fencing, we can write the equation as:
2(L + W) = 8
L + W = 4 (equation 1)
Next, let's calculate the area of the rectangular cage, which will be the sum of the areas of the three smaller cages:
Total Area = (L)(W/3) + (L)(W/3) + (L)(W/3)
Total Area = (LW) / 3 + (LW) / 3 + (LW) / 3
Total Area = (LW) / 3 (equation 2)
To find the maximum total enclosed area, we need to find the values of L and W that satisfy equations 1 and 2.
We can substitute the value of W from equation 1 into equation 2 to get a single variable equation in terms of L:
Total Area = (L)(4 - L) / 3
To find the maximum area, we take the derivative of the Total Area equation with respect to L and set it equal to zero:
d(Total Area) / dL = (4 - 2L) / 3 = 0
Solving for L, we get:
4 - 2L = 0
2L = 4
L = 2
Substituting this value of L back into equation 1, we can solve for W:
2 + W = 4
W = 2
Therefore, the overall dimensions that will yield the maximum total enclosed area are L = 2 meters and W = 2 meters.
In conclusion, the dimensions of the rectangular cage that divide it into three smaller cages and yield the maximum total enclosed area are 2 meters by 2 meters.