Answer:
Length (L) = 3x
Explanation:
Let's break down the problem step by step.
Let's assume the original dimensions of the rectangular piece of metal are:
Length = L
Width = W
According to the problem, when the longer side (length) is increased by 3 times (3x) and the shorter side (width) is increased by 2 meters (2m), the new dimensions become:
New length = L + 3x
New width = W + 2
We are also told that the new dimensions result in an area that is double the original area.
Step 1: Calculate the original area
Original area = Length × Width = L × W
Step 2: Calculate the new area
New area = New length × New width = (L + 3x) × (W + 2)
According to the problem, the new area is double the original area, so we can set up the equation:
2 × Original area = New area
Substituting the values, we have:
2 × (L × W) = (L + 3x) × (W + 2)
Step 3: Simplify the equation
Expand the equation:
2LW = LW + 2L + 3xW + 6x
Combine like terms:
LW - 2L - 3xW - 6x = 0
Now, let's rearrange the equation to group the variables together:
LW - 3xW - 2L - 6x = 0
Factoring out common terms:
W(L - 3x) - 2(L + 3x) = 0
At this point, we can see two terms multiplied by (L - 3x) and (L + 3x). We set each of these factors to zero:
L - 3x = 0
L + 3x = 0
Solving these equations individually, we get:
L = 3x
L = -3x
Since we are dealing with the dimensions of a physical object, negative values don't make sense in this context. Therefore, we can ignore the solution L = -3x.
Hence, the dimensions of the given piece of metal are:
Length (L) = 3x
Width (W) = Any value
Please note that the width can have any value as long as the length is three times the value of x.