A farmer needs to enclose three sides of a field with a fence (the fourth side is a river). The farmer has 32 feet of fence and wants the field to have an area of 128 sq-feet. What should the dimensions of the field 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.)
Well, that's a tricky one! Let's figure it out!
Let's say the width of the field is w feet. Since we need to enclose three sides, we need a total of 2w + 1l feet of fence.
According to the problem, the farmer has 32 feet of fence. So we can write the equation: 2w + l = 32.
Additionally, we know that the area of the field is 128 sq-feet. To calculate the area, we multiply the width by the length: w * l = 128.
Now, let's use a little comedic magic to solve this math puzzle!
Why don't we imagine the farmer wearing a clown suit while measuring the width of the field? That would definitely make it more amusing!
Now, let's calculate the possible dimensions of the field.
Since we want to maximize the length, let's start by assuming the width is 1 foot (after all, clowns like to perform in tight spaces!). So, 1 * l = 128, which means the length would be 128 feet!
However, since we need 32 feet of fence and have already used up 1 foot for the width, that leaves us with 31 feet for the remaining two sides. Using some clown math, we can express this as 2 * 1 + 1 * l = 31.
If we rearrange the equation, we get l = 31 - 2 = 29 feet.
So, with a width of 1 foot and a length of 29 feet, we have a clown-approved solution!
Remember, though, this is just one possibility. There might be other dimensions that could work too. But hey, who doesn't love a clown-themed field?
Let's denote the width of the field as 'w' and the length as 'l'.
We know that the field has three sides enclosed by a fence, so the perimeter of the field can be expressed as:
Perimeter = 2w + l
Given that the farmer has 32 feet of fence, we can set up the equation:
2w + l = 32
We also know that the area of the field is 128 square feet. The area of a rectangle can be calculated as:
Area = length * width
So we have the equation:
Area = wl = 128
To find the dimensions of the field, we can solve this system of equations.
First, let's solve the equation 2w + l = 32 for l:
l = 32 - 2w
Now substitute this value of l into the equation for the area:
wl = 128
w(32 - 2w) = 128
32w - 2w^2 = 128
2w^2 - 32w + 128 = 0
Divide through by 2 to simplify the equation:
w^2 - 16w + 64 = 0
This equation can be factored as:
(w - 8)^2 = 0
So we have only one solution for w:
w = 8 feet
Now substitute this value of w back into the equation 2w + l = 32 to solve for l:
2(8) + l = 32
16 + l = 32
l = 32 - 16
l = 16 feet
Therefore, the dimensions of the field should be 16 feet by 8 feet.
To solve this problem, we can use the formula for the area of a rectangle: Area = Length x Width. Given that the farmer wants the field to have an area of 128 sq-feet, we can write the equation as:
128 = Length x Width
Next, since the farmer has 32 feet of fence, we know that the total length of all three sides of the fence (excluding the river side) is 32 feet. In other words, we have:
Length + Width + Length = 32 feet
Simplifying this equation, we get:
2 Length + Width = 32 feet
Now, we have a system of two equations:
128 = Length x Width
2 Length + Width = 32 feet
To solve this system, we can use substitution or elimination. Let's use substitution:
From the second equation, we can express the value of Width in terms of Length:
Width = 32 - 2 Length
Now, substitute this value of Width into the first equation:
128 = Length x (32 - 2 Length)
Expanding the brackets:
128 = 32 Length - 2 Length^2
Rearranging the equation:
2 Length^2 - 32 Length + 128 = 0
Factoring out a common factor of 2:
2 (Length^2 - 16 Length + 64) = 0
Simplifying:
Length^2 - 16 Length + 64 = 0
This equation can be factored as:
(Length - 8)^2 = 0
Taking the square root:
Length - 8 = 0
Therefore, Length = 8 feet.
Now, substituting this value back into the equation for Width:
Width = 32 - 2 Length
Width = 32 - 2(8)
Width = 32 - 16
Width = 16 feet
So, the dimensions of the field should be 8 feet by 16 feet.
x = width perpendicular to river
y = length parallel to river
fence length = 2 x + y = 32
x y = 128 so y = 128 / x
then
2 x + 128/x = 32
2 x^2 - 32 x + 128 = 0
x^2 - 16 x + 64 = 0
(x-8)(x-8) = 0
x = 8
then y = 128/x = 16