Please show all of your work in the space below. Please present and organized and complete solution.
A garden will be made up of a parallelogram, a rectangle and a triangle. The garden must have an area of 500m square. Draw two possible gardens. Determine the dimensions of each part of the garden, and justify your choice of these dimensions.
To find two possible gardens with different dimensions that satisfy the given conditions, we need to consider the properties of a parallelogram, rectangle, and triangle.
Let's start by setting up equations for the area of each component of the garden:
1) Parallelogram:
The area of a parallelogram is given by the formula A = base × height.
Let's call the base of the parallelogram "b" and the height "h1".
2) Rectangle:
The area of a rectangle is also given by the formula A = length × width.
Let's call the length of the rectangle "l" and the width "w".
3) Triangle:
The area of a triangle is given by the formula A = (base × height) / 2.
Let's call the base of the triangle "b" (which is the same as the base of the parallelogram) and the height "h2".
We are given that the garden must have an area of 500m²:
A = 500m²
To find two possible solutions, we can assign different values to the variables while maintaining the overall area of the garden.
Let's consider the following dimensions for the first garden:
Parallelogram:
Base (b) = 20m
Height (h1) = 25m (A = b × h1 = 20m × 25m = 500m²)
Rectangle:
Length (l) = 10m
Width (w) = 50m (A = l × w = 10m × 50m = 500m²)
Triangle:
Base (b) = 20m (same as the base of the parallelogram)
Height (h2) = 50m (A = (b × h2) / 2 = (20m × 50m) / 2 = 500m²)
So, the first possible garden will have a parallelogram with a base of 20m and a height of 25m, a rectangle with a length of 10m and a width of 50m, and a triangle with a base of 20m and a height of 50m.
Now, let's consider different dimensions for the second garden:
Parallelogram:
Base (b) = 10m
Height (h1) = 50m (A = b × h1 = 10m × 50m = 500m²)
Rectangle:
Length (l) = 25m
Width (w) = 20m (A = l × w = 25m × 20m = 500m²)
Triangle:
Base (b) = 10m (same as the base of the parallelogram)
Height (h2) = 100m (A = (b × h2) / 2 = (10m × 100m) / 2 = 500m²)
So, the second possible garden will have a parallelogram with a base of 10m and a height of 50m, a rectangle with a length of 25m and a width of 20m, and a triangle with a base of 10m and a height of 100m.
In conclusion, we have found two possible gardens with different dimensions and justified our choice of dimensions by maintaining the overall area of the garden as 500m².