, AA′ = 33 m and BC =7.5 m. The span is divided into six equal parts at E, G,
C, I, and K. Find the length of A′B
To find the length of A'B, we need to determine the distance between points A' and B. Here's how you can do that:
1. Calculate the total span AB by adding the lengths of all the segments: AB = AE + EG + GC + CI + IK + KB.
2. Since the span is divided into six equal parts, each segment has the same length. Let's represent the length of each segment as x. So, AE = EG = GC = CI = IK = KB = x.
3. Since we know that BC = 7.5 m, we can rewrite GC as 2x (since GC + CI = BC).
4. With this information, we can write the equation for the total span AB: AB = x + x + 2x + x + x + x = 6x.
5. We also know that AA' = 33 m. Since AE = x, we can say A'A = 33 - x.
6. Now, we have a right triangle A'GC with sides A'G and GC. Using the Pythagorean theorem, we can write the equation: A'A^2 = AG^2 + GC^2. Rearranging, we have (33 - x)^2 = x^2 + (2x)^2.
7. Expanding and simplifying the equation will give us a quadratic equation: 1089 - 66x + x^2 = 5x^2.
8. Rearranging the equation, we get 4x^2 - 66x + 1089 = 0.
9. Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, using the quadratic formula is simpler: x = (-(-66) ± √((-66)^2 - 4(4)(1089))) / (2(4)).
10. Calculate x using the quadratic formula, and you'll find two possible values for x. Only one of these values will be applicable in the context of the problem.
11. Once you have the value of x, substitute it back into the equation AB = 6x to find the length of A'B.
To find the length of A'B, we first need to find the length of AB.
Since the span is divided into 6 equal parts, we can divide the length AA' (33 m) by 6 to find the length of each part.
Length of each part = AA' / Number of parts = 33 m / 6 = 5.5 m
Now, we can calculate the length of AB by adding the lengths of each part:
Length of AB = 5.5 m + 5.5 m + 5.5 m + 5.5 m + 5.5 m + 7.5 m
Length of AB = 35.5 m
Since A'B is the same length as AB, the length of A'B is also 35.5 m.