Captain salman has 12m of fencing that he wants to use to make rectangular pens to put around his garden to make predators out. He wants to enclose the biggest possible region. What is the area he can enclosed by using the given pens
A square is the largest area.
To find the area that Captain Salman can enclose using the given fencing, we need to determine the dimensions of the rectangular pens that would maximize the enclosed area.
Let's denote the length of one side of the rectangular pen as "l" and the length of the other side as "w." Since there are four sides to a rectangular pen, and Captain Salman has 12 meters of fencing, we can set up the following equation:
2l + 2w = 12
This equation represents the perimeter of the rectangular pen, which is equal to the total length of fencing that Captain Salman has.
Now, we can solve this equation for one variable and substitute it back into the area formula. Solving the equation for l, we get:
2l = 12 - 2w
l = 6 - w/2
The formula for the area of a rectangle is given by A = l * w. Substituting the expression for l into the area formula, we have:
A = (6 - w/2) * w
Expanding the equation, we get:
A = 6w - (w^2)/2
To find the maximum area, we need to find the value of w that maximizes this equation. One way to do this is by taking the derivative of the area equation with respect to w, setting it equal to zero, and solving for w.
dA/dw = 6 - w
Setting dA/dw equal to zero, we have:
6 - w = 0
Solving for w, we find:
w = 6
Therefore, the width of the rectangular pen that maximizes the enclosed area is 6 meters. We can substitute this value back into the area formula to find the maximum area:
A = (6 - 6/2) * 6
A = (6 - 3) * 6
A = 3 * 6
A = 18 square meters
Hence, Captain Salman can enclose an area of 18 square meters using the given 12 meters of fencing.