joans rectangular garden is 6 meters long and 4 meters wide she plans to double the area of the garden by increasing each side by the same amount find the number of meters by which each dimension must be increased
To double the area of the rectangular garden, we need to find the amount by which each dimension must be increased.
Let's start by finding the current area of the garden. The formula for finding the area of a rectangle is given by length multiplied by width.
Area = Length × Width
Given:
Length = 6 meters
Width = 4 meters
Current Area = 6 meters × 4 meters
Current Area = 24 square meters
To double the area, we need to increase the length and width by the same amount.
Let's assume the increase in length and width as 'x' meters.
The new length would be (6 + x) meters.
The new width would be (4 + x) meters.
The new area of the garden would be:
New Area = (6 + x) meters × (4 + x) meters
To double the area, the new area must be twice the current area:
New Area = 2 × Current Area
Substituting the values, we get:
(6 + x) meters × (4 + x) meters = 2 × 24 square meters
Expanding the equation:
(6 + x)(4 + x) = 48
Simplifying further:
24 + 4x + 6x + x^2 = 48
Combining like terms:
x^2 + 10x + 24 = 48
Rearranging the equation to bring all terms to one side:
x^2 + 10x + 24 - 48 = 0
x^2 + 10x - 24 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or quadratic formula to find the value of 'x'.
To double the area of the garden, we need to find out by how much each dimension needs to be increased.
First, let's find the current area of the garden. The formula for the area of a rectangle is length multiplied by width. In this case, the length is 6 meters and the width is 4 meters:
Area of the current garden = 6 meters * 4 meters = 24 square meters
To double the area, we need the new area to be twice the current area, or 2 * 24 square meters = 48 square meters.
Since both dimensions, length and width, need to be increased by the same amount, let's call this amount "x" meters.
The new length will be the current length (6 meters) plus "x" meters, and the new width will be the current width (4 meters) plus "x" meters.
The new area, with the increased dimensions, will be the new length multiplied by the new width.
Using the new length and width in the area formula, we can write the equation:
New area = (6 meters + x) * (4 meters + x)
We want the new area to be 48 square meters, so we can set up the equation:
(6 meters + x) * (4 meters + x) = 48
Now, let's solve this equation to find the value of "x" which represents the increase in each dimension.