Explanation Consider w be the width as a function of r . w = c 0 + c 1 r (1) Sketch the triangular cross section as shown in Figure 1. Refer to Figure 1. The width w = b at r = r 1 . The width w = 0 at r = r 2 . The height of the triangle is h = r 2 − r 1 . Calculate the width b 1 as shown below. Substitute b for w and r 1 for r in Equation (1). b = c 0 + c 1 r 1 (2) Similarly, Substitute 0 for w and r 2 for r in Equation (1). 0 = c 0 + c 1 r 2 (3) Reduce Equations (1) and (2) as shown below. b = c 0 + c 1 r 1 − ( c 0 + c 1 r 2 ) b = c 1 ( r 1 − r 2 ) c 1 = b − ( r 2 − r 1 ) Substitute h for r 2 − r 1 . c 1 = − b h Calculate the value of c 0 as shown below. Substitute − b ( r 2 − r 1 ) for c 1 in Equation (2). b = c 0 − b ( r 2 − r 1 ) r 1 b ( r 2 − r 1 ) = c 0 ( r 2 − r 1 ) − b r 1 b r 2 − b r 1 = c 0 ( r 2 − r 1 ) − b r 1 r 2 b = c 0 ( r 2 − r 1 ) c 0 = r 2 b r 2 − r 1 Substitute h for r 2 − r 1 . c 0 = r 2 b h Sketch the cross section of triangle with a strip of thickness dr as shown in Figure 2. Refer to Figure 2. Calculate the area of the strip ( d A ) as shown below. d A = w d r Substitute c 0 + c 1 r for w . d A = ( c 0 + c 1 r ) d r d A r = ( c 0 + c 1 r ) d r r Integrate both sides of the Equation

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Chapter 4.10, Problem 189P

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The expression of R for a triangular cross section.

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