A three-sided fence is to be built next to a straight section of a river, which forms the fourth side of a rectangular region. The enclosed area is to equal 3200 m2. What is the minimum perimeter that such a fence can have?
I developed two equations: 3600=xy
& 2x + y = P .... Now what????
You need to get one variable:
3600/y = x.
Then insert it into the perimeter equation:
2 (3600/y) + y = P (this way, you only have to solve for 1 variable)
Solve for Y. Substitue the Y value into the first equation of (3600=xy) to find X.
Once you have X and Y, solve for P.
To find the minimum perimeter for the fence, we need to minimize the value of P in the equation 2x + y = P.
First, let's substitute the value of x from the equation 3600 = xy into the perimeter equation:
2 (3600/y) + y = P
We now have a perimeter equation in terms of only y. To find the minimum value of P, we need to find the value of y that makes the equation minimum.
To do this, take the derivative of the equation with respect to y, set it equal to zero, and solve for y.
dP/dy = -7200/y^2 + 1 = 0
Solving for y, we get:
7200/y^2 = 1
y^2 = 7200
y = sqrt(7200)
Now that we have the value of y, we can substitute it into the equation 3600 = xy to solve for x.
3600 = x * sqrt(7200)
x = 3600 / sqrt(7200)
Finally, we can calculate the minimum perimeter by substituting the values of x and y into the perimeter equation:
P = 2 * x + y
P = 2 * (3600 / sqrt(7200)) + sqrt(7200)
Simplifying this expression will give you the minimum perimeter that the three-sided fence can have.