A farmer needs to enclose three sides of a pasture with a fence (the fourth side is a river). The farmer has 42 meters of fence and wants the pasture to have an area of 220 sq-meters. What should the dimensions of the pasture be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)

If the width is w, then the length is 42-2w. So, the area is

y = w(42-2w)
y = -2w^2 + 42w

this is just a parabola, and y reaches its maximum when w = -42/-4 = 10.5

so, the pasture is 10.5 x 21. Area = 220.5

If we want just 220 m^2, then the field is 11x20

Well, well, well, we've got a farmer with a tricky fencing problem! Let's put on our thinking caps and solve this one.

Alright, so we have three sides to fence and one side, which is conveniently a river. Total length of the fencing is 42 meters. Looks like we need to find the dimensions of the pasture.

Let's call the width of the pasture "w" and the length "l." Since we have three sides to fence, we can write the equation as follows:

2w + l = 42

Since the area of the pasture is given as 220 sq-meters, we can write another equation:

w * l = 220

Now, let's solve these equations like a detective solving a crime!

From the first equation, we can express l in terms of w:

l = 42 - 2w

Now substitute this into the second equation:

w * (42 - 2w) = 220

Expanding the equation and rearranging it, we get a quadratic equation:

-2w² + 42w - 220 = 0

Solving this equation with the quadratic formula, we find two possible values for w. But remember, we want the width to be the smaller dimension, so let's take the smaller value.

After some mathematical magic, we find that w ≈ 6 meters.

Now substitute this value of w back into the first equation to find l:

2 * 6 + l = 42
12 + l = 42
l = 42 - 12
l ≈ 30 meters

So, the dimensions of the pasture should be approximately 6 meters wide and 30 meters long. Don't forget to thank me for solving this puzzle of yours!

Let's assume the width of the pasture is 'w' meters and the length is 'l' meters.

Given that the area of the pasture is 220 square meters, we know that:

w * l = 220

The farmer needs to enclose three sides of the pasture with a fence, so the total perimeter of the pasture will be:

Perimeter = w + 2l

Given that the farmer has 42 meters of fence, we have:

w + 2l = 42

Now, we have a system of two equations:
1. w * l = 220
2. w + 2l = 42

We can solve this system of equations to find the values of 'w' and 'l'.

First, let's rearrange equation 2 to express 'w' in terms of 'l':

w = 42 - 2l

Now, substitute this value of 'w' into equation 1:

(42 - 2l) * l = 220

We can simplify equation 3:

42l - 2l^2 = 220

Rearrange equation 3 to form a quadratic equation:

2l^2 - 42l + 220 = 0

Let's solve this quadratic equation using factoring, completing the square, or using the quadratic formula:

2l^2 - 44l + 4l + 220 = 0
2l(l - 22) + 4(l - 22) = 0
(2l + 4)(l - 22) = 0

From this equation, we can see that either:

2l + 4 = 0, which gives l = -2 (not meaningful in this context)
or
l - 22 = 0, which gives l = 22

Since the length cannot be negative, we disregard the l = -2 solution.

Thus, the length is 22 meters.

Substitute this value of 'l' back into equation 2 to find 'w':

w + 2(22) = 42
w + 44 = 42
w = 42 - 44
w = -2 (not meaningful in this context)

Since the width cannot be negative, there is no valid solution in this case.

Therefore, there are no dimensions that satisfy the given conditions.

To solve this problem, we need to find the dimensions of the pasture that allows the farmer to enclose three sides with a fence using a total of 42 meters.

Let's assume the width of the pasture is "w" meters, and the length is "l" meters.

According to the given information, the pasture has an area of 220 sq-meters. So, we can write the equation:

w * l = 220

We also know that three sides of the pasture need to be fenced. Since the fourth side is a river, we only need to fence the width (w) twice and the length (l) once. The total fencing required is 42 meters. We can write the equation for the perimeter of the pasture as:

2w + l = 42

To find the dimensions of the pasture, we can use these two equations.

1. Rewrite the equation for the area in terms of w:
l = 220 / w

2. Substitute this value of l into the equation for the perimeter:
2w + (220 / w) = 42

3. Solve for w by rearranging the equation into quadratic form:
2w^2 + 220 - 42w = 0

4. Simplify the equation:
2w^2 -42w + 220 = 0

5. Solve this quadratic equation using factoring, completing the square, or the quadratic formula.

After solving the equation, you will find two possible values for w. Since the problem states that the length should be as long as possible, you will choose the larger value of w.

After finding the value of w, you can substitute it back into the equation l = 220 / w to find the length (l).

That will give you the dimensions of the pasture.