A gardener wants his garden to have a geometrical shape.he decides on the following rules for building the flower beds.1)They must all be rectangular.
2)The perimeter and area must be the same.How many different flower beds can the gardener make 1)if one of the side is 3 unit less than the other side.2)if both the sides are of same length.
1)1-1 and 4 (i can only think of this)
2)1-2 and 2
(1) 2(x + x-3) = x(x-3)
x = 1 or 6
But x=1 isn't viable, so only x=6
(2) 2(x+x) = x^2
x = 4
To find the number of different flower beds, we need to consider the given rules, analyze the conditions, and apply mathematical principles.
1) If one side is 3 units less than the other side:
Let's assume one side of the rectangle is x units. According to the condition, the other side will be (x + 3) units.
a) Perimeter: The perimeter of a rectangle is given by the formula P = 2 * (length + width). In this case, it would be P = 2 * (x + (x + 3)) = 2 * (2x + 3) = 4x + 6.
b) Area: The area of a rectangle is given by the formula A = length * width. In this case, it would be A = x * (x + 3) = x^2 + 3x.
In order to determine the number of different flower beds satisfying the given conditions, we need to find the values of x for which the perimeter and area are equal.
Setting the equations equal to each other:
4x + 6 = x^2 + 3x
Rearranging:
x^2 - x - 6 = 0
Factoring or using the quadratic formula, we find:
(x - 3)(x + 2) = 0
Therefore, x can be either 3 or -2. However, we cannot have a negative length for a flower bed. Thus, the only possible value is x = 3. Therefore, one side of the rectangle will be 3 units, and the other side will be (3 + 3) = 6 units.
Therefore, when one side is 3 units less than the other side, the gardener can make only one flower bed.
2) If both sides are of the same length:
Let's assume both sides of the rectangle are x units each.
a) Perimeter: P = 2 * (length + width). In this case, it would be P = 2 * (x + x) = 4x.
b) Area: A = length * width. In this case, it would be A = x * x = x^2.
For the perimeter and area to be equal:
4x = x^2
Rearranging:
x^2 - 4x = 0
Factoring or using the quadratic formula, we find:
x(x - 4) = 0
Therefore, x can be 0 or 4. But for a flower bed to exist, the length cannot be 0. Thus, the only possible value is x = 4.
Therefore, when both sides are of the same length, the gardener can make only one flower bed.
To summarize:
1) If one side is 3 units less than the other side, the gardener can make one flower bed with sides measuring 3 units and 6 units.
2) If both sides are of the same length, the gardener can make one flower bed with sides measuring 4 units each.