If you have 280 meters of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose?
Area =
140x70
maximum area is when the fencing is divided equally between lengths and widths.
Where's the solution?
To find the largest area you can enclose with a given amount of fencing, you need to determine the dimensions of the rectangle that will maximize the area.
Let's say the length of the side of the rectangle parallel to the wall is x meters. Since the fence will be up against the wall, this side will not require any fencing. Therefore, you have two equal lengths of fence that need to be used for the other three sides, totaling 280 meters.
Let's denote the width of the rectangle as y meters. Then, the total length of the three sides that require fencing is 2y. From the given information, we have:
2x + 2y = 280
Simplifying, we can divide through by 2:
x + y = 140
We can rearrange this equation to solve for y:
y = 140 - x
Now, to find the area of the rectangle, we use the formula:
Area = length × width
Area = x × (140 - x)
To find the value of x that maximizes the area, we need to find the vertex of the parabola representing the area function. This occurs at the x-coordinate of the vertex, which is given by:
x = -b / (2a)
For the parabola representing the area function, a = -1 and b = 140. Substituting these values, we get:
x = -140 / (2 × -1)
x = -140 / -2
x = 70
Now that we have the value of x, we can substitute it back into the equation for y:
y = 140 - x
y = 140 - 70
y = 70
Therefore, the dimensions of the rectangle that maximize the area are x = 70 meters and y = 70 meters. Substituting these values into the area formula:
Area = x × y
Area = 70 × 70
Area = 4,900 square meters
So, the largest area you can enclose with 280 meters of fencing is 4,900 square meters.
To find the largest area you can enclose with a given amount of fencing, you need to understand that the maximum area is obtained when you have a square shape. In this case, since the rectangle is up against a long, straight wall, the shape will be a rectangle with one side equal to the length of the wall.
Let's assume the length of the wall is L, and the width of the rectangle is W. Since the fence should enclose all four sides, we have:
Perimeter = 2L + W = 280 meters (equation 1)
To find the largest area, we need to express the area in terms of a single variable. Since we already have an equation for the perimeter, we can solve equation 1 for W:
W = 280 - 2L
Now, the area of a rectangle is given by the formula:
Area = Length × Width
Substituting the value of W from equation 1 into the area formula:
Area = L × (280 - 2L)
Expanding the expression:
Area = 280L - 2L²
Now, we have the area (A) expressed as a quadratic equation in terms of L. The largest area can be determined by finding the maximum value of the quadratic equation.
To find the maximum value of a quadratic equation of the form ax² + bx + c, where a < 0, we can use the formula:
x = -b / (2a)
In our case, a = -2 and b = 280. Plugging these values into the formula:
L = -280 / (2 × -2)
L = 140 / 2
L = 70
Therefore, the length of the rectangle that provides the largest area is 70 meters. Now, we can substitute this value into equation 1 to find the width:
W = 280 - 2L
W = 280 - 2(70)
W = 280 - 140
W = 140
Thus, the width of the rectangle is also 140 meters.
Finally, to find the area, we multiply the length (L) by the width (W):
Area = 70 × 140
Area = 9800 square meters.
Therefore, with 280 meters of fencing, you can enclose the largest area of 9800 square meters by creating a rectangular shape with dimensions 70 meters by 140 meters.