100m of fencing wire is available for making a rectangular enclosure.find the maximum area possible

Answer 1

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.