100m of fencing wire is available for making a rectangular enclosure.find the maximum area possible
To find the maximum area possible, we need to determine the dimensions of the rectangular enclosure that will use up all the fencing wire.
Let's assume that the rectangle has a length of x and a width of y. The perimeter of the rectangle, which is the total length of fencing wire needed, is:
P = 2x + 2y
We know that the total length of fencing wire available is 100m, so we can set up the equation:
2x + 2y = 100
Simplifying this equation, we get:
x + y = 50
Now we want to maximize the area, which is given by:
A = xy
We can solve for one variable in terms of the other using the perimeter equation. For example, we can solve for y:
y = 50 - x
Substituting this into the area equation, we get:
A = x(50 - x)
Expanding the expression, we get:
A = 50x - x^2
To find the maximum area, we need to find the value of x that maximizes this expression. We can do this using calculus. Taking the derivative of A with respect to x, we get:
dA/dx = 50 - 2x
Setting this equal to zero to find the critical point, we get:
50 - 2x = 0
x = 25
This critical point corresponds to a maximum value of A, since the second derivative of A is negative (i.e. the graph of A is concave down at x = 25). Therefore, the dimensions of the rectangular enclosure that will maximize the area are:
x = 25m (length)
y = 25m (width)
And the maximum area possible is:
A = xy = 25m x 25m = 625m^2
Therefore, a rectangular enclosure with length 25m and width 25m will have the maximum area possible with 100m of fencing wire.