The bridge will span a section that is 35 meters wide, with ends of the bridge set back 2.5 m from each river bank. The engineer must apply right angle relationships because one river bank is 9 m higher than the other.
What is the final length of this bridge?
To find the final length of the bridge, we need to calculate the diagonal length of the right-angled triangle formed by the width of the section and the height difference between the river banks.
Step 1: Calculate the length of the hypotenuse (diagonal):
Using the Pythagorean theorem, we can find the length of the hypotenuse (diagonal) of the right-angled triangle.
c^2 = a^2 + b^2
Given:
Width of the section (base of the triangle) = 35 m
Height difference between river banks (height of the triangle) = 9 m
Here, one side (a) is the width of the section (35 m), and the other side (b) is the height difference between the river banks (9 m).
Plugging in the values into the formula, we get:
c^2 = (35)^2 + (9)^2
c^2 = 1225 + 81
c^2 = 1306
Step 2: Find the square root of both sides to solve for c:
c = √1306
c ≈ 36.11 meters (rounded to two decimal places)
So, the final length of the bridge is approximately 36.11 meters.
To find the final length of the bridge, we can break down the problem into two components: the horizontal distance (span) and the vertical distance (height difference).
1. Horizontal distance (span):
The bridge spans a section that is 35 meters wide, with ends set back 2.5 meters from each river bank. Since both ends are set back, we need to subtract the set back distance from the total span.
Span = 35 meters - 2.5 meters - 2.5 meters = 30 meters
2. Vertical distance (height difference):
One river bank is 9 meters higher than the other. To find the vertical distance between the two ends of the bridge, we need to use right angle relationships. We can consider the height difference as the length of one side of a right triangle, while the distance across the river is the hypotenuse.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse (final length of the bridge):
length^2 = span^2 + height^2
length^2 = 30 meters^2 + 9 meters^2
length^2 = 900 meters^2 + 81 meters^2
length^2 = 981 meters^2
length ≈ √981 meters
length ≈ 31.29 meters
So, the final length of the bridge is approximately 31.29 meters.
so the total horizontal span of the bridge is 40m, with one end 9m higher.
The final length z of the bridge is thus
z^2 = 40^2 + 9^2