a rectangular field is to be enclosed with 240 m of fence. find a mathematical model expressing the area of the field as a function of its length.
Ah, the mysterious rectangular field! It's like watching a shape-shifting magician in action. Now, let's figure out this mathematical riddle and find the hidden treasure.
Let's call the length of the field "L" and the width "W". To find the area, we need a formula that relates the length and width with the perimeter. Well, guess what? I happen to know the secrets!
Starting with the perimeter, we have 240 m = 2L + 2W. Since there are two lengths and two widths (remember the magic trick?), each counted twice due to the fencing.
Now, my friend, let's isolate one variable and express it in terms of the other—like a complex dance! Let's solve the equation above for W:
240 m = 2L + 2W (dividing both sides by 2)
120 m = L + W (subtracting L from both sides)
W = 120 m - L (voila!)
Finally, let's express the area in terms of the length (L) and width (W). Since Area = Length × Width, we can substitute the value of W we just found:
Area = L × (120 m - L)
There you have it! The mathematical model expressing the area of the field as a function of its length:
Area = L × (120 m - L)
Now, go forth, brave mathematician, and uncover the hidden treasures of rectangular fields!
Let's assume the length of the rectangular field is 'l' and the width is 'w'.
For a rectangular field, the perimeter is given by the formula:
Perimeter = 2(length + width)
We are given that the perimeter of the field is 240m. So, we can write the equation as:
240 = 2(l + w)
Next, we want to express the area of the field as a function of its length. The area of a rectangle is given by the formula:
Area = length * width
Since we are expressing the area as a function of the length, we need to eliminate the width (w) using the perimeter equation.
Rearranging the perimeter equation, we have:
l + w = 120
Substituting the value of w = 120 - l in the area formula, we get:
Area = length * (120 - length)
Therefore, the mathematical model expressing the area of the field as a function of its length is:
A(l) = l * (120 - l)
To find a mathematical model that expresses the area of the field as a function of its length, we need to use the given information about the fence.
Let's assume the length of the rectangular field is "L" meters. Since the field is rectangular, we know that there are two sides with length L and two sides with width W.
The perimeter of the field is given as 240 m, which means the total length of the fence used is 240 m. From this information, we can form the equation:
Perimeter = 2L + 2W
Substituting the given perimeter value:
240 = 2L + 2W
We are looking for the area of the field, which is determined by multiplying the length and width:
Area = L * W
To express the area as a function of the length, we need to eliminate the width variable from the equation.
From the perimeter equation, we can solve for W:
240 - 2L = 2W
Dividing both sides by 2:
120 - L = W
Now, substitute this value for W in the area equation:
Area = L * (120 - L)
Simplifying:
Area = 120L - L²
Therefore, the mathematical model expressing the area of the field as a function of its length is given by the equation:
Area(L) = 120L - L²
2(width+length)=240
width+length = 120
width = 120-length
area = length*width = length*(120-length)