a regular plot of land is designed so that its length is 6 meters more than its width. the diagonal of the land is 10 meters. to the nearest tenth of a meter, what are the dimensions of the land?
d^2 = x^2 + (x+6)^2 = 10^2
x^2 + x^2 + 12x + 36-100=0
x^2 + 6x -32 = 0
completing the square:
x^2 + 6x + 9 = 32+9
(x+3)^2 = 41
x + 3 = √41 , I ignored the negative answer
x = √41-3 = appr 3.403
one side is 3.403 , the other is 9.403
check:
3.403^2 + 9.403^2
= 99.996818
diagona = √... =9.99984
good enough
To find the dimensions of the land, we can use the Pythagorean theorem, which relates the sides of a right triangle. In this case, the length and width form two sides of a right triangle, and the diagonal is the hypotenuse.
Let's denote the width of the land as 'w'. According to the problem, the length is 6 meters more than the width, so the length can be represented as 'w + 6'.
Using the Pythagorean theorem, we have:
(diagonal)^2 = (width)^2 + (length)^2
Replacing the variables with their values, we get:
10^2 = w^2 + (w + 6)^2
Now, we can solve this equation to find the width of the land.
Expanding and simplifying the equation, we have:
100 = w^2 + (w^2 + 12w + 36)
Combining like terms:
100 = 2w^2 + 12w + 36
Rearranging the equation to form a quadratic equation:
2w^2 + 12w + 36 - 100 = 0
2w^2 + 12w - 64 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Using the quadratic formula, we have:
w = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 2, b = 12, and c = -64. Substituting these values, we get:
w = (-12 ± √(12^2 - 4(2)(-64))) / (2(2))
Simplifying the equation, we have:
w = (-12 ± √(144 + 512)) / 4
w = (-12 ± √656) / 4
w = (-12 ± 25.61) / 4
Now, we can find the two possible values for the width 'w':
w₁ = (-12 + 25.61) / 4 ≈ 3.65
w₂ = (-12 - 25.61) / 4 ≈ -9.16
Since we cannot have a negative width for the land, we conclude that the width is approximately 3.65 meters.
To find the length, we substitute this value back into the equation we used earlier:
length = w + 6
length = 3.65 + 6
length ≈ 9.65 meters
Therefore, the dimensions of the land are approximately 3.65 meters by 9.65 meters.
Let's solve this problem step-by-step:
Step 1: Assign variables to the unknowns.
Let's call the width of the land "w" (in meters).
Step 2: Translate the given information into equations.
We are told that the length of the land is 6 meters more than its width, so the length can be represented as "w + 6".
We also know that the diagonal of the land is 10 meters. Using the Pythagorean theorem, we can relate the width, length, and diagonal of a right-angled triangle formed by the width, length, and diagonal. The equation is as follows:
width squared + length squared = diagonal squared
w^2 + (w + 6)^2 = 10^2
Step 3: Solve the equation.
Expanding (w + 6)^2, we get:
w^2 + w^2 + 12w + 36 = 100
Combining like terms:
2w^2 + 12w + 36 - 100 = 0
2w^2 + 12w - 64 = 0
Divide the equation by 2 to simplify:
w^2 + 6w - 32 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's factor it.
(w + 8)(w - 4) = 0
Setting each factor equal to zero:
w + 8 = 0 or w - 4 = 0
Solving for "w":
w = -8 or w = 4
Since measurement cannot be negative, we discard the negative value.
Therefore, the width of the land is 4 meters.
Step 4: Calculate the length.
The length is given by "w + 6".
Substituting the value of "w":
length = 4 + 6 = 10 meters
Step 5: Verify the solution using the Pythagorean theorem.
width squared + length squared = diagonal squared
4^2 + 10^2 = 100
16 + 100 = 100
116 = 100
The equation does not hold true for the given dimensions, which means there may be an error in the initial problem statement.
Please double-check the information provided or reconfirm the problem's details.