Final answer:
To solve the problem, two equations were set up based on the fencing and area requirements: 2W + L = 110 meters and L * W = 1200 square meters. Solving this system revealed that the width of the field is 20 meters and the length is 60 meters to meet both criteria.
Step-by-step explanation:
The student is asking to solve a mathematical problem related to finding the dimensions of a rectangular field that a farmer wants to enclose with a fence. Since one side of the field is along a river and does not need fencing, the length of the fence the farmer has, which is 110 meters, will be used for the other three sides of the rectangle. Furthermore, the field needs to have an area of 1200 square meters.
Let's denote the length of the side parallel to the river as L and the width of the field (the two sides that will be fenced) as W. Since there's no fence on the river side, the total length used for fencing is 2W + L = 110 meters. We also have the area of the field, which is L * W = 1200 square meters.
To find the dimensions L and W, we need to solve this system of equations:
2W + L = 110 (total length of the fence)
L * W = 1200 (area of the field)
From the first equation, we can express L as L = 110 - 2W. Substituting this into the second equation gives us:
W*(110 - 2W) = 1200
Expanding and rearranging the equation, we have:
2W2 - 110W + 1200 = 0
Dividing the entire equation by 2 to simplify, we get:
W2 - 55W + 600 = 0
This is a quadratic equation which we can solve by factoring or by using the quadratic formula. Factoring, we find:
(W - 20)(W - 30) = 0
Therefore, W can be either 20 meters or 30 meters. If W is 20 meters, then L is 110 - 2*20 = 70 meters. If W is 30 meters, then L is 110 - 2*30 = 50 meters.
However, since we're looking to fence a rectangle and already have a predefined area, we can see that the dimensions that satisfy the area requirement of 1200 square meters are W = 20 meters and L = 60 meters