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.
No there is no common side (lets say.)
Thanks
To express the area of the enclosures as a function of its dimensions, we need to consider the separate rectangular areas for the sheep and cattle enclosures.
Let's assume the dimensions of the sheep enclosure are length (L1) and width (W1), and the dimensions of the cattle enclosure are length (L2) and width (W2).
Now, let's calculate the amount of fencing needed for each enclosure.
For the sheep enclosure, we need two lengths and two widths, so the total fencing needed is:
2L1 + 2W1.
Similarly, for the cattle enclosure, we need two lengths and two widths, so the total fencing needed is:
2L2 + 2W2.
Given that there is 240m of fencing available, we can set up the equation:
2L1 + 2W1 + 2L2 + 2W2 = 240.
Now, we can solve this equation for one variable in terms of the other. Let's solve for W1:
2W1 = 240 - 2L1 - 2L2 - 2W2,
W1 = 120 - L1 - L2 - W2.
To find the area of the sheep enclosure, we multiply the length (L1) by the width (W1):
Area of the sheep enclosure (A1) = L1 * W1 = L1 * (120 - L1 - L2 - W2).
Similarly, the area of the cattle enclosure (A2) is given by:
Area of the cattle enclosure (A2) = L2 * (120 - L1 - L2 - W2).
So, the area of the enclosures as a function of their dimensions is:
A1 = L1 * (120 - L1 - L2 - W2),
A2 = L2 * (120 - L1 - L2 - W2).
This equation represents the area of each enclosure in terms of their dimensions, given the total amount of fencing available.