A garden has a rectangular pool in it with a width that is twice its length. If the land around the pool is 4 meters on all sides and the total area of the land is 136 m^2, what are the dimensions of the pool?
if the dimensions of the pool are w and 2w, then
(w+8)(2w+8) - w(2w) = 136
To solve this problem, we can follow these steps:
Step 1: Assign variables:
Let's assign the variable "x" to represent the length of the pool.
Since the width is twice the length, we can assign the variable "2x" to represent the width of the pool.
Step 2: Determine the dimensions of the land:
From the given information, we know that the land area surrounding the pool is 4 meters on all sides. Therefore, the length of the land would be (x + 4 + 4) meters, and the width would be ((2x) + 4 + 4) meters.
Step 3: Calculate the area of the land:
The area of the land can be found by multiplying the length and width:
Area of the land = (length of the land) * (width of the land) = (x + 8) * (2x + 8)
According to the problem, the area of the land is 136 m^2, so we can set up the equation:
(x + 8) * (2x + 8) = 136
Step 4: Solve the equation:
Expand the equation: 2x^2 + 8x + 16x + 64 = 136
Combine like terms: 2x^2 + 24x + 64 = 136
Rearrange the equation: 2x^2 + 24x - 72 = 0
Divide through by 2: x^2 + 12x - 36 = 0
Step 5: Factor or use the quadratic formula to solve for x:
This quadratic equation can be factored as: (x + 6)(x - 6) = 0
Setting each factor equal to zero gives:
x + 6 = 0 or x - 6 = 0
Solving each equation gives:
x = -6 or x = 6
Since we are dealing with dimensions, a negative value for x is not feasible.
Thus, x = 6, which represents the length of the pool.
Step 6: Calculate the width of the pool:
The width of the pool is twice its length, so the width = 2x = 2 * 6 = 12 meters.
Therefore, the dimensions of the pool are:
Length = 6 meters
Width = 12 meters