Denzel wants to rope off a 800 m2
rectangular
swimming area using the beach as one of the
sides. What should the dimensions of the
rectangle be in order to use the minimum
amount of rope?
So you need 2 widths and 1 length
2w + l = 800
l = 800-2w
area = lw =w(800-2w)
= -2w^2 + 800w
a downward parabola, with a min value at its vertex.
the w of the vertex is -800/-4 = 200 ,
( in y = ax^2 + bx + c, the x of the vertex is -b/(2a) )
so the area is 200 m by 800-2(200) or 400 m
Well, Denzel certainly doesn't want to make any waves here, so let's come up with a solution that's tide-y. If Denzel wants to use the minimum amount of rope, he should create a square swimming area since a square has equal sides.
So, a square with an area of 800 m² would have sides of approximately √800 ≈ 28.3 m. Therefore, the dimensions of the rectangle should be 28.3 m by 28.3 m to keep things perfectly afloat while minimizing the rope needed.
To minimize the amount of rope required, the rectangle should be as square as possible.
Since one of the sides of the rectangle will be the beach, we can assume that the length of this side is 800 meters. Let's call the other side of the rectangle x meters.
The formula to calculate the perimeter of a rectangle is P = 2L + 2W, where P is the perimeter, L is the length, and W is the width.
In this case, we want to minimize P by making the rectangle as square as possible, so we want the length and width to be as close as possible.
Since the beach side is 800 meters, the perimeter equation becomes:
P = 2(800) + 2x = 1600 + 2x.
To make the rectangle as square as possible, we want the length and width to be equal, so x = 800/2 = 400 meters.
Therefore, the dimensions of the rectangle should be 800 meters by 400 meters in order to use the minimum amount of rope.
To find the dimensions of the rectangle that will use the minimum amount of rope, we need to consider that one side of the rectangle will be the beach. Let's assume the length of the beach side is "x".
The other two sides of the rectangle will form the remaining sides of the swimming area. Let's call these sides "y" and "z".
Since we want to minimize the amount of rope needed, we need to minimize the perimeter of the rectangle. The perimeter is equal to the sum of all four sides of the rectangle.
The perimeter can be calculated using the formula:
perimeter = x + y + x + z
Given that the total area of the rectangle is 800 m², we can also use the formula for the area of a rectangle:
area = length * width
Substituting the variables:
800 = x * y
Rearranging the equation:
y = 800 / x
Now substitute the value of "y" into the perimeter equation:
perimeter = x + (800 / x) + x + z
perimeter = 2x + (800 / x) + z
To minimize the amount of rope, we need to minimize the perimeter. That means we need to find the minimum value of the expression 2x + (800 / x) + z.
One approach to find the minimum value is to use calculus by taking the derivative of the expression with respect to "x". However, this would require more information about the relationship between "x" and "z". Since we don't have that information, we can proceed with optimization techniques.
A common approach in optimization problems is to solve for one variable in terms of the other(s), and then substitute into the expression to eliminate that variable.
From the area equation:
800 = x * y
y = 800 / x
Substituting this value of "y" into the perimeter equation:
perimeter = 2x + (800 / x) + z
perimeter = 2x + (800 / x) + z
Now we can see that the perimeter only depends on "x" and "z". We need to minimize it, but without further information about the relationship between "x" and "z", we cannot calculate the exact minimum dimensions.
However, we can explore some possible scenarios to get a better understanding:
- If z = 0, then the rectangle becomes a line segment and the perimeter reduces to 2x.
- If x = z, then the rectangle becomes a square and the perimeter reduces to 4x.
In these scenarios, the minimum perimeter occurs when:
- z = 0, and x = 400 m. The dimensions of the rectangle are then 400 m x 0 m.
- x = z, and x = sqrt(800) = 28.28 m. The dimensions of the rectangle are then 28.28 m x 28.28 m.
These are just two examples, but without more information about the relationship between "x" and "z", we cannot determine the exact minimum dimensions.