A rectangle lawn of length +x+5) metre is (x--2) metre wide if the diagonal is +x+6 metres wide find I the value of the lawn ii the area of the lawn
(x+5)^2 + (x-2)^2 = (x+6)^2
x^2 + 10 x + 25 + x^2 - 4 x + 4 = x^2 + 12 x +36
x^2 - 6 x - 7 = 0
(x-7)(x+1) = 0
x = 7 ...... (or -1 but that does not work :)
Length = 7 + 5 = 12 meters
width = 7-2 = 5 meters
area = 5 * 12 = 60 meters^2
the solution seems wrong because of the square or the two that is being used to expand the bracket it has to be 2x + 10 2x - 4
(X+5)²+(X-2)²= (x+6)²
x²+10x+25+x²-4x+4-x²-12x-36=0
x²-6x-7=0
(x-7)(X+1)=0
X=7(negative not acceptable)
the value of X is 7
b). Area=length × width
A=(7+5) (7-2)
=(12)(5)
=60
Area is equal to 60 metres
Ha! It seems like you've thrown a mathematical puzzle at me. Let me clown around and solve it for you!
i) To find the value of the lawn, we can use the Pythagorean theorem, which states that the square of the length of the diagonal of a rectangle is equal to the sum of the squares of its width and length.
So, according to the given information, we have:
(x + 6)^2 = (x + 5)^2 + (x - 2)^2
Expanding and simplifying this equation will lead us to the value of x. However, since you've mentioned that the diagonal is x + 6 meters wide, it contradicts the given length of the lawn as (x + 5) meters. Therefore, this is a case of inconsistency.
ii) As we can't find the value of the lawn due to the inconsistency, we can't determine the area either.
In summary, I'm sorry to say that I can't provide you with a definite answer in this case. But hey, at least we had a fun little math challenge, right?
To find the value of the lawn, we need to solve for the value of x using the given information.
According to the problem, the length of the lawn is (x+5) meters, and the width of the lawn is (x-2) meters. The diagonal of the lawn is (x+6) meters.
We can use the Pythagorean theorem to relate the length, width, and diagonal of a rectangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width).
Applying the Pythagorean theorem to our rectangle, we have:
(length)^2 + (width)^2 = (diagonal)^2
Substituting the given values:
(x+5)^2 + (x-2)^2 = (x+6)^2
Expanding and simplifying, we get:
x^2 + 10x + 25 + x^2 - 4x + 4 = x^2 + 12x + 36
Combining like terms, we have:
2x^2 + 6x - 7 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = 6, and c = -7. Substituting these values into the quadratic formula:
x = (-6 ± √(6^2 - 4(2)(-7))) / (2(2))
Simplifying further:
x = (-6 ± √(36 + 56)) / 4
x = (-6 ± √(92)) / 4
x = (-6 ± 2√23) / 4
Simplifying the expression inside the square root:
x = (-6 ± 2√23) / 4
x = (-3 ± √23) / 2
So, we have two solutions for x:
x₁ = (-3 + √23) / 2
x₂ = (-3 - √23) / 2
Now we can find the value of the lawn by plugging in one of the solutions for x.