A garden in the farm of a square, is surrounded by a path which is 1m wide on two opposite sides and 1/2m wide on the other two opposite sides. If the area of the path is 17msquared, calculate the :
Perimeter of the garden
Area of the space covered by the garden and the path
side of the square --- x m
length of garden = x + 2 m , 1 m on each end
width of garden = x + 1 m
area of path = (x+2)(x+1) - x^2 = 17
x^2 + 3x + 2 - x^2 = 17
carry on
Why -x^2?
I took the whole garden: (x+2)(x+1)
and subtracted the inner part, which is x^2,
leaving me with the area of the path around the inner garden.
And of course that was given as 17, thus my equation.
Thank you very much Reiny. Done it. :) :)
To solve this problem, we can break it down into smaller steps.
Step 1: Find the dimensions of the garden
Let's assume the sides of the garden are x meters each. Since the path is 1m wide on two opposite sides and 1/2m wide on the other two opposite sides, the dimensions of the garden would be (x + 1 + 1) and (x + 1/2 + 1/2).
So, the dimensions of the garden are (x + 2) and (x + 1).
Step 2: Calculate the area of the path
The area of the path can be calculated by subtracting the area of the garden from the total area of the garden and path. Since the total area of the path and garden is given as 17m², we can set up the equation:
(x + 2)(x + 1) - x^2 = 17
Expanding and simplifying this equation, we get:
x^2 + 3x + 2 - x^2 = 17
3x + 2 = 17
3x = 15
x = 5
So, the side length of the garden is 5 meters.
Step 3: Calculate the perimeter of the garden
Since the garden is square, the perimeter is simply 4 times the length of one side. In this case, the length of one side is 5 meters, so the perimeter of the garden is:
4 * 5 = 20 meters
Step 4: Calculate the area of the space covered by the garden and the path
The area of the garden can be found by squaring the side length, which is 5 meters in this case:
Area of garden = 5 * 5 = 25 m²
The area of the space covered by the garden and the path is the sum of the area of the garden and the area of the path:
Area of garden and path = 25 + 17 = 42 m²
Therefore, the perimeter of the garden is 20 meters and the area of the space covered by the garden and the path is 42 square meters.