Questions: A large pipe is held in place by using a 2 -ft-high block on one side and a 5 - ft -high block on the other side. If the length of the arc between the points where the pipe touches the blocks is 14 ft , then what is the radius of the pipe? Hint: Ignore the thickness of the pipe.

A large pipe is held in place by using a 2 -ft-high block on one side and a 5 - ft -high block on the other side. If the length of the arc between the points where the pipe touches the blocks is 14 ft , then what is the radius of the pipe?
Hint: Ignore the thickness of the pipe.

Transcript text: A large pipe is held in place by using a 2 -ft-high block on one side and a 5 - ft -high block on the other side. If the length of the arc between the points where the pipe touches the blocks is 14 ft , then what is the radius of the pipe? Hint: Ignore the thickness of the pipe.

Solution

Solution Steps

Step 1: Calculate Height Difference

The height difference between the two blocks is given by: \[ \text{height\_diff} = \text{height2} - \text{height1} = 5 - 2 = 3 \text{ ft} \]

Step 2: Establish Relationships

Using the relationship between the arc length \( s \), the radius \( r \), and the central angle \( \theta \) in radians, we have: \[ s = r \theta \] where \( s = 14 \text{ ft} \).

Additionally, the height difference can be expressed in terms of the radius and the central angle: \[ \text{height\_diff} = r \left(1 - \cos\left(\frac{\theta}{2}\right)\right) \]

Step 3: Solve for the Radius

By solving the equations simultaneously, we find that the radius \( r \) is approximately: \[ r \approx 7.6063 \text{ ft} \]