A rectangular flower garden, 7 m by 6 m, is surrounded by a grass strip of uniform width. If the total area of the garden and the grass strip is 90 m^2, what is the width of the strip?
To solve this, I used the formula 90=(6+2x)(7+2x) where is is the width of one ised, but since it's surrounded on all sides, I used 2x. The problem is, I'm getting a negative answer; the back of the books tells me I should be getting 1.5m as an answer, but I'm clearly doing something wrong. I just don't know what though and would really appreciate help understanding my errors.
4 x^2 + 26 x + 42 = 90
4 x^2 + 26 x - 48 = 0
2 x^2 + 13 x - 24 = 0
(2 x - 3)( x + 8) = 0
2x = 3 or x = -8
x = 3/2 or x = -8
use x = 3/2 = 1.5
To determine the width of the grass strip, we can set up the equation by considering the total area of the garden and the grass strip.
Let's represent the width of the grass strip as 'x'. Since the garden is surrounded on all sides by the grass strip, the dimensions of the garden including the grass strip will be (7+2x) meters by (6+2x) meters.
The total area of the garden and the grass strip is given as 90 m^2. Therefore, we can set up the equation as:
(7+2x)(6+2x) = 90
Expanding this equation, we get:
42 + 14x + 12x + 4x^2 = 90
Combining like terms, we have:
4x^2 + 26x + 42 = 90
Rearranging the equation, we get:
4x^2 + 26x + 42 - 90 = 0
4x^2 + 26x - 48 = 0
Now, we can solve this quadratic equation for the value of 'x'.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 4, b = 26, and c = -48, substituting these values into the quadratic formula gives:
x = (-26 ± √(26^2 - 4(4)(-48))) / (2(4))
Simplifying the equation further:
x = (-26 ± √(676 + 768)) / 8
x = (-26 ± √1444) / 8
x = (-26 ± 38) / 8
Now we have two possible solutions for 'x':
x1 = (-26 + 38) / 8 = 12 / 8 = 1.5 (width cannot be negative)
x2 = (-26 - 38) / 8 = -64 / 8 = -8
Since the width cannot be negative, we discard the solution x = -8.
Therefore, the width of the grass strip is 1.5 meters.
To find the width of the grass strip, let's walk through the problem step by step.
We know that the total area of the garden and the grass strip is 90 m^2. The garden itself has dimensions of 7 m by 6 m.
To calculate the area of the entire rectangular region, including the garden and the grass strip, the width of the grass strip needs to be added to the length and width of the garden.
We can represent the width of the grass strip as 'x', so the length of the entire region would be (7+2x) and the width would be (6+2x).
The area of a rectangle is found by multiplying its length by its width. Therefore, the equation can be written as:
90 = (7+2x)(6+2x)
To solve this equation, we can expand it by using the distributive property:
90 = 42 +14x + 12x + 4x^2
Combining like terms, it becomes:
4x^2 + 26x + 42 = 90
Rearranging this equation to form a quadratic equation:
4x^2 + 26x + 42 - 90 = 0
4x^2 + 26x - 48 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
Factoring the quadratic equation:
4x^2 + 26x - 48 = 0
(2x + 12)(2x - 4) = 0
Setting each factor equal to zero:
2x + 12 = 0 or 2x - 4 = 0
Solving each equation:
2x + 12 = 0
2x = -12
x = -6 (which is a negative value)
2x - 4 = 0
2x = 4
x = 2
We obtained two potential values for 'x', one negative and one positive. However, in this context, we are dealing with the width of a strip, which cannot be negative. Therefore, we discard the -6 and conclude that the width of the grass strip is:
x = 2 m
So, the width of the strip is 2 m, not 1.5 m as mentioned in the back of the book. It seems there might be an error in the answer key.