Let's assume the width of the container is "x" meters.
According to the given information, the length of the container is one meter longer than the width, so the length would be "x + 1" meters.
The height is stated to be one meter greater than twice the width, which can be expressed as "2x + 1" meters.
To find the dimensions of the container, we need to consider the volume formula for a rectangular solid:
Volume = Length × Width × Height
Plugging in the given volume of 84 m^3 and the dimensions we determined:
84 = (x + 1) × x × (2x + 1)
Expanding and simplifying the equation:
84 = (2x^2 + x) × (x + 1)
84 = 2x^3 + 3x^2 + x
Rearranging the equation and setting it equal to zero:
2x^3 + 3x^2 + x - 84 = 0
Now, we can use numerical methods or a graphing calculator to solve this equation. After solving it, we find that one possible solution is x ≈ 3.35.
Since the dimensions of a container cannot be negative, we discard any negative solutions.
Therefore, the width of the container is approximately 3.35 meters. Using this value, we can find the length and height:
Length = x + 1 ≈ 3.35 + 1 ≈ 4.35 meters
Height = 2x + 1 ≈ 2(3.35) + 1 ≈ 7.7 meters
So, the dimensions of the container should be approximately:
Width ≈ 3.35 meters
Length ≈ 4.35 meters
Height ≈ 7.7 meters