A rectangular garden plot 16 by 24 meters is to be bordered by a strip of uniform width x meter so as to double the area. Find x.
current area = 16(24) m^2 = 384 m^2
new plot is 16+2x by 24+2x
new area = (16+2x)(24+2x) = 2(384)
expand, simplify to a quadratic equation.
Solve this quadratic using the method you learned.
hint: it factors and has a nice integer solution
To double the area of the rectangular garden plot, we need to add a strip of uniform width around the border.
Let's consider the dimensions of the new rectangle including the border:
Length of the new rectangle = Length of the original rectangle + 2 * Width of the border
Width of the new rectangle = Width of the original rectangle + 2 * Width of the border
Given that the original rectangular garden plot measures 16 by 24 meters, we can substitute these values to form the equations.
Length of the new rectangle = 16 + 2x
Width of the new rectangle = 24 + 2x
The area of the new rectangle is double the area of the original rectangle. So, we can set up the equation:
Area of the new rectangle = 2 * Area of the original rectangle
(16 + 2x) * (24 + 2x) = 2 * 16 * 24
Expanding the equation:
384 + 48x + 32x + 4x^2 = 768
Simplifying the equation:
4x^2 + 80x - 384 = 0
Dividing the equation by 4 to simplify it further:
x^2 + 20x - 96 = 0
Now, we can solve this quadratic equation to find the value of x.
Factoring the equation:
(x + 24)(x - 4) = 0
Setting each factor equal to zero and solving for x:
x + 24 = 0 or x - 4 = 0
If x + 24 = 0, then x = -24, which is not a valid solution in this context, as the width cannot be negative.
If x - 4 = 0, then x = 4. This is the valid solution.
Therefore, the width of the border strip (x) is 4 meters.
To find the width of the border strip, we need to determine the original area of the rectangular garden plot.
The original area of the rectangular garden plot is given by the formula:
Area = length × width
Given that the length is 24 meters and the width is 16 meters, the original area is:
Original Area = 24 × 16 = 384 square meters
To double the area, we need to add a border strip around the garden plot. Since the width of the strip is uniform, it will be added on all four sides, increasing the length and width of the garden plot by 2x meters.
The new length of the garden plot will be 24 + 2x meters, and the new width will be 16 + 2x meters.
The new area, after adding the border strip, will be given by:
New Area = (24 + 2x) × (16 + 2x)
Since we want to double the original area, the new area should be 2 times the original area. Therefore, we can set up the equation:
2 × Original Area = New Area
Substituting the values, we get:
2 × 384 = (24 + 2x) × (16 + 2x)
Simplifying the equation:
768 = (24 + 2x)(16 + 2x)
Expanding:
768 = 384 + 48x + 32x + 4x^2
Rearranging and combining like terms:
4x^2 + 80x + 384 - 768 = 0
4x^2 + 80x - 384 = 0
Dividing the equation by 4:
x^2 + 20x - 96 = 0
Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula.
Factoring the equation, we get:
(x + 24)(x - 4) = 0
Setting each factor equal to zero and solving for x, we get:
x + 24 = 0 or x - 4 = 0
x = -24 or x = 4
Since the width cannot be negative, the width of the border strip is 4 meters.
Therefore, the width of the border strip is 4 meters.