a gardner having 150 meters of fencing materials wishes to enclose a rectangular plot of land and also erect a fence across the land parallel to two of the sides.find the maximum area he can enclose
where is the solution
To find the maximum area that can be enclosed, we need to determine the dimensions of the rectangular plot that will maximize the area.
Let's assume the length of the rectangular plot is "l" in meters, and the width is "w" in meters.
Since the perimeter of the rectangular plot is 150 meters, we can form an equation:
2(l + w) = 150
Simplifying the equation, we have:
l + w = 75
Now, we need to express the area in terms of one variable to find the maximum. The area of a rectangle is given by:
A = l * w
We can rewrite it as:
A = w(75 - w)
To find the maximum area, we need to find the value of "w" that maximizes this function. One way to do this is by graphing the function, but let's use calculus.
To find the maximum, we will differentiate the function with respect to "w" and set it equal to zero:
dA/dw = 75 - 2w = 0
Solving this equation, we get:
2w = 75
w = 37.5
Next, substitute the value of "w" back into the equation l + w = 75:
l + 37.5 = 75
l = 37.5
So, the length and width of the rectangular plot that will maximize the area are 37.5 meters and 37.5 meters.
Now, to find the maximum area, substitute the values of "l" and "w" into the area formula:
A = l * w = 37.5 * 37.5 = 1406.25 square meters
Therefore, the maximum area that can be enclosed is 1406.25 square meters.
To find the maximum area that can be enclosed with a given amount of fencing materials, we need to determine the dimensions of the rectangular plot that would result in the largest area.
Let's assume the width of the rectangular plot is x meters. The length of the plot would then be (150 - 2x) meters to account for the fencing materials used for the width and the two parallel sides.
The area of a rectangle is given by the formula: Area = length * width.
Substituting the length and width values into the formula, we get:
Area = (150 - 2x) * x
Area = 150x - 2x^2
To find the maximum area, we need to maximize the quadratic equation. This can be done by finding the vertex of the quadratic equation, which represents the maximum point.
The x-coordinate of the vertex of a quadratic equation in the form of ax^2 + bx + c is given by: x = -b / 2a.
In our case, the quadratic equation is -2x^2 + 150x.
Plugging in the values, we have:
x = -150 / (2 * -2)
x = 75 / 2
x = 37.5
Since we cannot have a decimal value for the width of the plot, we can round down to the nearest whole number. Thus, the width (x) would be 37 meters.
The length of the rectangular plot would then be:
Length = 150 - 2x
Length = 150 - 2 * 37
Length = 150 - 74
Length = 76 meters
Therefore, with 150 meters of fencing materials, the gardener can enclose a rectangular plot with dimensions of 37 meters by 76 meters, resulting in a maximum area of 2,812 square meters.