The width of a rectangular carpet is 7 meters shorter than the length, and the diagonal is 1 meter longer than the length. What are the carpet's dimensions?
length -- x
width --- x-7
hypotenuse -- x+1
x^2 + (x-7)^2 = (x+1)^2
x^2 + x^2 - 14x + 49 = x^2 + 2x + 1
x^2 - 16x + 48=0
(x-4)(x-12) = 0
x = 4 or x=12 , but x > 7 to have a positive width
so x = 12
length = 12 metres
width = 5 metres
Well, it seems like this rectangular carpet has a very dramatic relationship going on. Let's try to figure it out!
Let's call the length of the carpet "L". According to the problem, the width of the carpet is 7 meters shorter than the length, so it would amount to "L - 7".
Now, we're told that the diagonal of the carpet is 1 meter longer than the length. So, the diagonal would be "L + 1".
Now, let's use some good old Pythagoras' theorem to solve this mystery. The square of the length plus the square of the width should be equal to the square of the diagonal.
So, we have (L^2) + (L - 7)^2 = (L + 1)^2. Let's simplify that!
L^2 + L^2 - 14L + 49 = L^2 + 2L + 1.
Simplifying further, we get L^2 - 16L + 48 = 0.
Let's solve this quadratic equation. Drumroll, please...
Using some clown magic known as factoring, we find that L = 12 or L = 4.
Well, since the width is 7 meters shorter than the length, if L is 12, then the width would be 5 (12 - 7), but if L is 4, then the width would be -3 (4 - 7).
Since negative widths are not very practical or logical for a carpet, we'll go with L = 12 and the width = 5.
So, the dimensions of the carpet are 12 meters by 5 meters.
And there you have it! The carpet's dimensions straight from the circus of humor!
Let's assume the length of the rectangular carpet is "x" meters.
According to the problem, the width of the carpet is 7 meters shorter than the length, which means the width is (x - 7) meters.
We are also given that the diagonal is 1 meter longer than the length. By using the Pythagorean theorem, we can find the diagonal of the rectangle in terms of its length and width.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Applying this to our rectangle, the diagonal, length, and width form a right triangle, where the diagonal is the hypotenuse.
So, according to the Pythagorean theorem, we have:
diagonal^2 = length^2 + width^2
(x + 1)^2 = x^2 + (x - 7)^2
Expanding the equation:
x^2 + 2x + 1 = x^2 + x^2 - 14x + 49
Combine like terms:
2x + 1 = 2x^2 - 14x + 49
Rearrange the terms:
2x^2 - 16x + 48 = 0
Divide the equation by 2:
x^2 - 8x + 24 = 0
To solve this quadratic equation, we can factorize it:
(x - 6)(x - 4) = 0
Setting each factor to zero:
x - 6 = 0, which gives us x = 6
x - 4 = 0, which gives us x = 4
So we have two possible values for x, which represent the length of the carpet - 6 and 4.
However, since the width should be shorter than the length, we can conclude that the length of the carpet is 6 meters.
Substituting this value into the equation for the width:
Width = x - 7 = 6 - 7 = -1 (Since width can't be negative, it means we made an error in our assumption)
Therefore, we made an error in our calculations as we can't have a negative width for the carpet.
Hence, there is no valid solution satisfying the given conditions.
Let's solve the problem step by step.
Let's assume the length of the rectangular carpet is "L" meters.
According to the problem, the width is 7 meters shorter than the length, so the width would be "L - 7" meters.
Now, we are given that the diagonal is 1 meter longer than the length. The diagonal of a rectangle forms a right triangle with the length and width as its two sides. We can use the Pythagorean theorem to find the diagonal.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width).
Using the theorem, we can express the diagonal as:
(diagonal)^2 = (length)^2 + (width)^2
Substituting the given values, we get:
(diagonal)^2 = L^2 + (L-7)^2
Expanding and simplifying the equation:
(diagonal)^2 = L^2 + L^2 - 14L + 49
Simplifying further:
(diagonal)^2 = 2L^2 - 14L + 49
Now, we are told that the diagonal is 1 meter longer than the length, so we can write the equation as:
(diagonal)^2 = L^2 + 1
Equating the two equations:
2L^2 - 14L + 49 = L^2 + 1
Rearranging the equation:
L^2 - 14L + 48 = 0
Now, we can solve this quadratic equation for "L" using factorization, completing the square, or quadratic formula.
Factoring the equation:
(L - 12)(L - 4) = 0
From this equation, we can see that either L - 12 = 0 or L - 4 = 0. Solving these equations:
Case 1: L - 12 = 0
L = 12
Case 2: L - 4 = 0
L = 4
Now, we have two possible values for the length. To find the corresponding values of the width, we can substitute these lengths into the expression: width = L - 7.
For L = 12, width = 12 - 7 = 5 meters.
For L = 4, width = 4 - 7 = -3 meters.
Since width cannot be negative, we disregard the solution L = 4.
Therefore, the dimensions of the carpet are length = 12 meters and width = 5 meters.