1.
A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. The enclosed area is to equal 1800 ft2. Find the minimum perimeter and the dimensions of the corresponding enclosure.
If side x is parallel to the river,
xy=1800
The perimeter is
p = 2(x+y) = 2(x + 1800/x)
p is a max when 1 - 1800/x^2 = 0, or x = 30√2
So, the field is 30√2 by 30√2
As usual, maximum area (or minimum perimeter) is achieved by a square.
To find the minimum perimeter and dimensions of the enclosure, we can start by setting up an equation using the given information.
Let's assume that one side of the rectangular region is x feet and the adjacent side is y feet, forming the rectangle.
Since the rectangle is surrounded by a three-sided fence, the perimeter would be the sum of the lengths of the three sides of the rectangle, which is 2x + y.
The area of the rectangle can be found by multiplying the length and width, which is xy.
According to the given information, the area of the rectangle should be 1800 ft^2. Therefore, we have the equation:
xy = 1800
To minimize the perimeter, we need to find the minimum value of 2x + y by taking the derivative with respect to x and setting it equal to zero. However, since we don't have any constraints on the dimensions, we can solve for y in terms of x using the equation xy = 1800 and substitute it into the perimeter equation.
Solving for y in terms of x:
y = 1800 / x
Substituting this into the perimeter equation:
Perimeter = 2x + (1800 / x)
To minimize the perimeter, we can take the derivative of the perimeter equation with respect to x:
d/dx (2x + (1800 / x)) = 2 - (1800 / x^2)
Setting the derivative equal to zero to find the critical points:
2 - (1800 / x^2) = 0
Simplifying:
2x^2 = 1800
Dividing by 2:
x^2 = 900
Taking the square root:
x = ±30
Since the dimensions cannot be negative, we take the positive value:
x = 30 ft
Substituting this value back into the equation xy = 1800 to solve for y:
30y = 1800
y = 1800 / 30
y = 60 ft
Therefore, the dimensions of the enclosure that would result in the minimum perimeter are 30 ft by 60 ft, and the corresponding perimeter is 2(30) + 60 = 120 ft.