Diagonal brace. The width of a rectangular gate is 2 meters
(m) larger than its height. The diagonal brace measures square root of 6. Find the width and height.
It's not going to be a very tall gate.
Let x=height of the gate.
Then x+2 = width of the gate.
Using pythagoras theorem,
x^sup2;+(x+2)² = 6
2x² + 4x - 2 = 0
Solve for x to get:
x=±√2 -1
x=0.41421 or -2.41421 (approx.)
The second root is rejected (height cannot be negative).
Therefore the height is approximately 0.41 m and the width is approx. 2.41 m.
So just to be clear, I wanted to make sure...the 6 is inside a square root box/sign..does this change ? Thx.
To find the width and height of the rectangular gate, we can set up a system of two equations based on the given information.
Let's denote the height of the gate as "h" meters. According to the problem, the width of the gate is 2 meters larger than its height, so the width can be represented as "h + 2" meters.
We are also given that the length of the diagonal brace measures the square root of 6. By using the Pythagorean theorem, we know that the square of the diagonal length is equal to the sum of the squares of the width and height. Therefore, we can write the equation:
(h + 2)^2 + h^2 = 6
Now, let's solve this equation to find the value of the height (h) and the width (h + 2).
Expanding the equation, we have:
h^2 + 4h + 4 + h^2 = 6
Combining like terms:
2h^2 + 4h + 4 = 6
Rearranging the terms:
2h^2 + 4h - 2 = 0
Divide both sides by 2 to simplify the equation:
h^2 + 2h - 1 = 0
To solve this quadratic equation, we can use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 2, and c = -1. Plugging in the values, we have:
h = (-2 ± √(2^2 - 4 * 1 * -1)) / (2 * 1)
Simplifying further:
h = (-2 ± √(4 + 4)) / 2
h = (-2 ± √8) / 2
h = (-2 ± 2√2) / 2
h = -1 ± √2
Since the height of a gate cannot be negative, we will consider the positive root:
h = -1 + √2
Now, substituting this value into the equation for the width:
w = h + 2
w = (-1 + √2) + 2
w = 1 + √2
Therefore, the width of the gate is 1 + √2 meters, and the height is -1 + √2 meters (approximately 0.414 meters).