Emma's rectangular garden is enclosed with 24 m of fencing. What is the least possible area of Emma's garden? What is the greatest possible area?
I don't understand what to do? How do we find the least and greatest possible area?
Lauren
Grade 6
P = 2(L + W)
P = 2 (6 + 6)
P = 2(5 + 7)
P = 2(4 + 8)
P = 2(3 + 9)
P = 2(2 + 10)
P = 2(1 + 11)
A = LW
A = 6 * 6
A = 5 * 7
Etc.
To find the least and greatest possible area of Emma's garden, we need to consider the dimensions of the garden.
Let's assume the length of the garden is L and the width is W.
Since the garden is rectangular, the perimeter of the garden is given by the formula:
Perimeter = 2L + 2W = 24 m
We can solve this equation for one variable and substitute it back into the area formula to calculate the least and greatest possible area.
Let's solve the equation for L:
2L + 2W = 24
2L = 24 - 2W
L = (24 - 2W) / 2
L = 12 - W
Now we can substitute this value of L into the area formula:
Area = L * W
Area = (12 - W) * W
Area = 12W - W^2
To find the greatest possible area, we need to maximize this equation. Since W^2 is always equal to or greater than 0, the maximum value of Area occurs when 12W is maximum, which is when W is highest possible value, i.e., W = 12/2 = 6.
Substituting W = 6 into the equation:
Area = 12W - W^2
Area = 12*6 - 6^2
Area = 72 - 36
Area = 36 sq.m
Therefore, the greatest possible area of Emma's garden is 36 square meters.
To find the least possible area, we need to minimize this equation. The lowest possible value for W is 0 since we can't have negative dimensions.
Substituting W = 0 into the equation:
Area = 12W - W^2
Area = 12*0 - 0^2
Area = 0
Therefore, the least possible area of Emma's garden is 0 square meters.
So, the least possible area is 0 sq.m and the greatest possible area is 36 sq.m.
To find the least and greatest possible area of Emma's garden, we need to consider the given information that the garden is enclosed with 24 m of fencing.
To find the least possible area, we can start by recognizing that a rectangle has opposite sides of equal length. Since the garden is enclosed with 24 m of fencing, we can divide the fence into two equal sides of length x and two equal sides of length y.
Therefore, the perimeter of the garden can be expressed as 2x + 2y = 24.
Simplifying the equation, we get x + y = 12.
To minimize the area, we can consider the extreme case when one side has a length of 0. For instance, if y is 0, then x must be 12 to satisfy the equation x + y = 12. This would result in a line segment, not a rectangle, so we discard this case.
Therefore, to find the least possible area, we need to find the dimensions of the rectangle that have the smallest possible difference between x and y.
Let's consider a common method for finding the maximum or minimum of a function using calculus. To minimize the difference between x and y, we can find the minimum value of their product xy. We can use calculus to find the minimum.
The area of the rectangle is A = xy.
To find the minimum value of A = xy, we can differentiate with respect to x and set the derivative equal to zero. Differentiating both sides of A = xy yields dA/dx = y.
Since x + y = 12, substitute y = 12 - x into dA/dx:
dA/dx = 12 - x.
To find the critical points, we set dA/dx equal to zero:
12 - x = 0.
Solving for x, we find x = 12.
Since x + y = 12, substitute x = 12 into the equation to find y:
12 + y = 12,
y = 0.
This means that x = 12 and y = 0 will result in a minimum area, which is a line segment, not a rectangle. It does not satisfy the requirements of a rectangle, so we discard this case.
Hence, the least possible area of Emma's rectangular garden is 0 square meters.
Now let's find the greatest possible area.
The greatest possible area of a rectangle given a fixed perimeter is achieved when the shape is a square. In this case, to maximize the area, the sides of the rectangle should be equal.
Since the perimeter is 24 m, we can set up the equation 2x + 2x = 24, where 2x represents the sum of all the sides of the rectangle.
Simplifying the equation, we get 4x = 24.
Solving for x, we find x = 6.
This means that both sides of the rectangle have a length of 6 m.
Therefore, the greatest possible area of Emma's rectangular garden is 6 m x 6 m = 36 square meters.
In summary, the least possible area is 0 square meters, and the greatest possible area is 36 square meters.