A 20-foot-long footbridge has two diagonal supports that meet in the center of the bridge. Each support makes a 65∘ angle with a short vertical support. What is the length x of a diagonal support, to the nearest tenth of a foot?
According to my interpretation, x would be the length of
the hypotenuse.
so cos 65° = 20/x
x = 20/cos65 = ......
12.5
Well, it seems like this footbridge is really trying to get its angles right! Let's calculate the length of the diagonal support, shall we?
Since the supports meet in the center of the bridge, they form an isosceles triangle. This means that the two diagonal supports are of equal length. Let's call that length of each support "x".
Now, we know that each diagonal support makes a 65-degree angle with a short vertical support. Since the two diagonal supports and the short vertical support are all part of a right triangle, we can use trigonometry to find the length of "x".
We can use the sine function, which relates the length of the opposite side (in this case, "x") to the hypotenuse (in this case, the length of the diagonal support).
sin(65) = x / 20
To solve for "x", we can rearrange the equation:
x = 20 * sin(65)
Calculating this out, we find that "x" is approximately equal to 18.7 feet (rounded to the nearest tenth of a foot).
So, the length "x" of each diagonal support is about 18.7 feet. Keep in mind, though, that this is just an approximation. So, don't forget to take these numbers with a grain of laughter!
To find the length of the diagonal support, we can use the concept of trigonometry. Let's call the length of the diagonal support x.
In a right triangle, we can use the trigonometric function tangent (tan) to relate the length of the opposite side (vertical support) to the length of the adjacent side (half of the bridge length).
Using the given information, we have:
tan(65°) = (opposite side) / (adjacent side)
We know that the adjacent side is half of the bridge length, which is 20 feet/2 = 10 feet.
So we have:
tan(65°) = x / 10
To solve for x, we can rearrange the equation:
x = 10 * tan(65°)
Using a calculator, we can find:
x ≈ 21.2 feet (to the nearest tenth of a foot)
Therefore, the length of the diagonal support is approximately 21.2 feet.
To find the length x of a diagonal support, we can use trigonometry and apply the sine function.
Let's break down the problem:
We are given that the two diagonal supports meet in the center of the footbridge, forming a right triangle with the footbridge itself.
One of the diagonal supports makes a 65∘ angle with a short vertical support, which forms another right triangle with the footbridge.
We want to find the length x, which represents the hypotenuse of the right triangle formed by the diagonal support and the footbridge.
To find x, we need to find the length of one of the legs of the right triangle. Specifically, we need to find the height of the footbridge.
Since the diagonal support forms a right triangle with the footbridge, we can use the sine function to relate the given angle, 65∘, with the height and the length of the footbridge.
The sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In this case, the opposite side is the height, and the hypotenuse is the length of the footbridge.
So we have:
sin(65∘) = height / 20 ft
To find the height, we rearrange the equation:
height = sin(65∘) * 20 ft
Now we can calculate the value of the height using a calculator:
height ≈ sin(65∘) * 20 ≈ 18.006 ft
Now that we have the height, we can use the Pythagorean theorem to find the length x.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In our case, one side is the height and the other side is half the length of the footbridge (since the two diagonal supports meet at the center).
Using the Pythagorean theorem:
x^2 = height^2 + (20 ft/2)^2
x^2 = 18.006^2 + 10^2
x^2 = 324.108036 + 100
x^2 = 424.108036
Now, we solve for x:
x ≈ √(424.108036)
x ≈ 20.6 ft
Therefore, the length x of a diagonal support is approximately 20.6 feet to the nearest tenth of a foot.