Geometry

https://copilot.microsoft.com/th/id/BCO.f08aa4ac-0da4-4931-81ab-a04d31c5b116.png h=0.22 r=0.15 angle in radians to two decimal places Then find the ctross sectional area inside the pipe above the water rounding final answer to 2 decimal places. show as much work as posssible

Step-by-step solution with explanation

Final Answer

Central angle of dry segment: **2.16 radians**. Cross-sectional area above the water (dry region): **0.02 m²** (precise value: 0.0150 m²).

Step-by-step solution

Identify the Circle Segment Setup

We have a circular pipe with radius r = 0.15 m. The water depth is h = 0.22 m. Since h = 0.22 > r = 0.15, the water is above the center, so the DRY (air) region is a circular segment at the TOP of the pipe. We need the cross-sectional area of that dry region above the water.

Find the Depth of the Dry Segment

The total pipe diameter is 2r = 0.30 m. The water fills 0.22 m from the bottom, so the dry air space at the top has a height (sagitta) of 0.08 m. This is the value of d we use for the dry circular segment.

Find the Half-Angle Using Sagitta Formula

For a circular segment, the relationship between the sagitta (segment height) and the half-angle θ at the center is: sagitta = r − r·cos θ, so cos θ = (r − sagitta)/r. Here cos θ = 0.07/0.15.

Calculate the Half-Angle θ in Radians

Taking the inverse cosine of 0.4667 gives us the half-angle of the dry segment measured from the top of the pipe down to the water surface chord. The full central angle of the dry segment is 2θ ≈ 2(1.0808) = 2.16 radians (to two decimal places).

State the Full Central Angle to Two Decimal Places

The central angle subtended by the dry segment (the arc above the water) is 2θ ≈ 2.16 radians. This is the angle asked for, rounded to two decimal places.

Apply the Circular Segment Area Formula

The area of a circular segment is A = (r²/2)(α − sin α), where α is the full central angle. We substitute r = 0.15 and α = 2θ = 2.1616 rad (using the unrounded value for accuracy).

Compute sin(2θ) and Plug In Values

We calculate sin(2.1616) ≈ 0.8242. Then we subtract: 2.1616 − 0.8242 = 1.3374. Multiplying r²/2 = 0.0225/2 = 0.01125 by 1.3374 gives the segment area.

Calculate the Final Dry Cross-Sectional Area

Multiplying gives A ≈ 0.01504 m². Rounded to 2 decimal places this is 0.02 m². To keep more practical precision: the cross-sectional area of the dry region inside the pipe above the water is approximately 0.02 m² (or more precisely 0.0150 m²).

Understanding this problem

Learning Insight

A circular segment is the 'slice' cut off by a chord. Its area depends on the central angle and radius via A = (r²/2)(α − sin α). The sin α term subtracts the triangular part, leaving only the curved cap. This formula comes from subtracting the isoceles triangle area from the sector area.

Quick Tip

Always check if the water depth h is above or below the center (r). If h > r, the DRY segment is the small cap on top, so use sagitta = 2r − h for the air space, not h itself.

Common Mistake

Students often use h = 0.22 directly as the sagitta for the segment formula, forgetting that h > r means the water is past the center. The correct sagitta for the DRY region is 2r − h = 0.08, not 0.22.