David has 160 metres of fencing to build a rectangular corral. He wants to give his animals as much space as possible by using the length of fencing that he can afford. Find the dimension and the maximum area of the corral.
Common sense will tell you that the enclosure would have to be a square,
so if each side is x, then 4x = 160
x = 40 m
area = 40^2 m^2 = 1600 m^2
but ....
let the width be x and the length be y m
then 2x+2y = 160
x+y = 80, or y = 80-x
area = xy = x(80-x) - -x^2 + 80x
this is a quadratic and would graph as a parabola
the x of the vertex is -b/(2a) = -80/-2 = 40
if x = 40, then y = 80-40 = 40
area = xy = 40*40 = 1600 m^2
(as anticipated above)
the largest area to perimeter ratio for a rectangle is a square
L + W = 80 ... L = 80 - W
A = L * W = 80 W - W^2
the max is on the axis of symmetry of the parabola
... Wmax = - b / (2 a) = -80 / (2 * -1) = 40
Lmax = 80 - Wmax = 40
Amax = Lmax * Wmax = 40 * 40 = 1600 m^2
To find the dimensions and maximum area of the corral, we need to use the given information about the amount of fencing David has.
Let's assume the length is "L" and the width is "W" of the rectangular corral.
We know that the perimeter of a rectangular corral is calculated by adding the lengths of all its sides. In this case, it would be:
Perimeter = 2L + 2W
Since David has 160 meters of fencing available, this can be expressed as:
2L + 2W = 160
Simplifying this equation, we can further express it as:
L + W = 80
Now we need to express one of the variables in terms of the other, so we can find the maximum area. Solving the equation above for L, we get:
L = 80 - W
The area (A) of a rectangular corral is given by multiplying its length and width. In this case, it would be:
A = L * W
Substituting the value of L from the equation above into the area formula, we get:
A = (80 - W) * W
Now, we need to find the value of W that maximizes the area. To do this, we can find the derivative of the area function and set it equal to zero.
dA/dW = 0
Differentiating the area function, we get:
dA/dW = 80 - 2W
Setting this equal to zero and solving for W, we get:
80 - 2W = 0
2W = 80
W = 40
So, the width W of the corral is 40 meters.
Substituting this value back into the equation L + W = 80, we get:
L + 40 = 80
L = 80 - 40
L = 40
Therefore, the dimensions of the corral are Length (L) = 40 meters and Width (W) = 40 meters.
To find the maximum area, we substitute these values into the area formula:
A = L * W
A = 40 * 40
A = 1600 square meters
So, the maximum area of the corral is 1600 square meters.