A rectangular lawn is surrounded on three sides by flower beds. Each flowerbed is 2m wide. The area of the lawn is 14m(squared). What is the length of the lawn?
<---------------- 3x---------------------->
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|. | /|\
| 2m _____________________|. |
|<------> |. |. |
|. |. LAWN |. 2x
|. |____________________|. |
|. |. |
|____________________________|. \|/
My Workings:
3x - 2 ----------------->
__________________
|. |
|. 14m(squared) |. <------- 2x - 4
|. |
|_________________|
3x - 2 x 2x - 4 (?)
How would you do this?
(3x-2)(2x-4) = 14
6x^2-16x-6 = 0
2(x-3)(2x+1) = 0
x = 3
So the bed is 7x2
But you could have guessed that at first, since 2 and 7 are the only factors of 14...
Well, this is a tricky one! But let me try to help you out with a little humor.
First, let's analyze the situation. We know that the lawn is surrounded on three sides by flower beds, which are each 2m wide. The area of the lawn is given as 14m².
Now, we can represent the length of the lawn as 3x-2, where x is the width of the lawn. The width of the lawn would be 2x-4, as we need to subtract the widths of the two flower beds on the sides.
To find the length of the lawn, we need to solve the equation that represents the area:
(3x-2) * (2x-4) = 14m²
Now, let's do some calculations and see where it leads us.
3x * 2x = 6x²
3x * -4 = -12x
-2 * 2x = -4x
-2 * -4 = 8
Putting it all together, we have:
6x² - 12x - 4x + 8 = 14m²
Simplifying a bit:
6x² - 16x + 8 = 14m²
Hmmm, looks like we have a quadratic equation on our hands. But fear not, my friend! We can solve this with a little bit of humor.
Let's convert the equation to a more comedic form, shall we?
6x² - 16x - 6 = 0
Now, we can use the quadratic formula to find the value of x.
But before I do that, let me just mention that I've always found it amusing how mathematicians make everything seem so serious. I mean, why can't they solve equations while wearing clown shoes? It would definitely make math class more entertaining!
Now, let's get back to business. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 6, b = -16, and c = -6. Plugging in the values, we get:
x = (-(-16) ± √((-16)² - 4*6*(-6))) / (2*6)
Simplifying that:
x = (16 ± √(256 + 144)) / 12
And if we keep our clown shoes on and do some further calculations, we get:
x = (16 ± √400) / 12
Which simplifies to:
x = (16 ± 20) / 12
Here comes a punchline!
If we choose the positive square root, x = (16 + 20) / 12 = 36 / 12 = 3.
So, the width of the lawn is 3m.
Now, let's put that back into our original equation for the length of the lawn:
3x - 2 = 3 * 3 - 2 = 9 - 2 = 7m.
Tada! The length of the lawn is 7m.
I hope that brought a smile to your face, even if it didn't make math any less confusing!
To find the length of the lawn, we need to set up an equation using the given information.
Let's assume the width of the lawn is "x" meters.
Since the lawn is surrounded by flower beds on three sides, the length of the lawn would be 2 meters less than three times the width.
So, the length of the lawn can be represented as (3x - 2).
We also know that the area of the lawn is 14 square meters.
The formula to find the area of a rectangle is length multiplied by width. Therefore, we can write the equation as:
length x width = area
(3x - 2) x x = 14
Now, let's solve this equation to find the value of "x" and subsequently calculate the length of the lawn.
3x^2 - 2x = 14
Rearranging the equation, we get:
3x^2 - 2x - 14 = 0
Now, we can solve this quadratic equation. There are a few methods to do this, such as factoring, completing the square, or using the quadratic formula. Let's solve it using factoring.
Factoring the equation, we can write it as:
(3x + 4)(x - 2) = 0
Setting each factor equal to zero, we have:
3x + 4 = 0, x - 2 = 0
Solving for "x":
3x = -4, x = 2
Since the width of the lawn cannot be negative, we discard the solution x = -4.
Therefore, the width of the lawn is x = 2 meters.
Substituting this value back into our initial equation for the length, we find:
Length of the lawn = (3x - 2) = (3 * 2 - 2) = 4 meters.
Hence, the length of the lawn is 4 meters.
To find the length of the lawn, we can follow these steps:
1. Start by identifying the given information:
- The lawn is a rectangular shape.
- It is surrounded on three sides by flower beds, each 2m wide.
- The area of the lawn is 14m².
2. Let's assume the width of the lawn is "x" meters. Since there are two flower beds on each side, the actual width of the whole rectangle (lawn + flower beds) would be (x + 2 + 2) = (x + 4) meters.
3. Since the length of the lawn is not given directly, we'll use the given area of 14m² to find the length.
4. Using the formula for the area of a rectangle (Area = Length x Width), we have:
Area = Length x Width
14m² = Length x (x + 4)m
5. Simplifying the equation:
14m² = Length x (x + 4)m
14m² = x² + 4x
6. Rearranging the equation to solve for "x" in terms of Length:
x² + 4x - 14 = 0
7. Factoring the quadratic equation or using the quadratic formula, we find two possible solutions for "x" (the width of the lawn). However, in this case, since the width cannot be negative, we can discard the negative solution. The positive solution is:
x = 2m
8. Now that we know the width of the lawn is 2m, we can find the length by adding the widths of the lawn and the flower beds:
Length = x + 4
Length = 2m + 4
Length = 6m
Therefore, the length of the lawn is 6m.