1) A rancher wants to enclose two rectangular areas near a river, one for sheep and one for cattle. There is 240m of fencing available. Express the area of the enclosures as a function of its dimension.
You have to be more specific to describe the rectangles.
In this type of question, there is usually some kind of common side, or sides that are not needed.
In your case, the river might make one side to the rectangle unnecessary.
But, ...
Is there a common side to the two rectangles. ???
....
That is the question, it can be solved in multiple ways from what I understand.
So , tell me which way you want it to be.
No there is no common side (lets say.)
Thanks by the way.
To express the area of the enclosures as a function of its dimension, let's assume that the dimensions of the rectangular areas are represented by length (L) and width (W).
For the sheep enclosure, we need four sides of fencing: two lengths and two widths. This gives us a perimeter of 2L + 2W. Since we have 240m of fencing available, we can set up the equation:
2L + 2W = 240
We can simplify this equation by dividing both sides by 2:
L + W = 120
Now, to find the area of the sheep enclosure, we multiply the length (L) by the width (W):
Area of Sheep Enclosure = L * W
Similarly, for the cattle enclosure, we would use the remaining fencing. Let's denote the length and width of the cattle enclosure as L' and W'. The perimeter equation for the cattle enclosure would be:
2L' + 2W' = 240
Simplifying this equation as we did before, we get:
L' + W' = 120
And the area of the cattle enclosure would be:
Area of Cattle Enclosure = L' * W'
So, the area of the enclosures can be expressed as a function of their dimensions as follows:
Area of Sheep Enclosure = L * W
Area of Cattle Enclosure = L' * W'