Explanation Given information: A square has side of 10 cm. Area of the triangle is 30 cm 2 . Calculation: Let the area of the triangle be A 1 . A 1 = 30 cm 2 (1) The expression for area of triangle is shown below: A = 1 2 b h Here, the width of the triangle is b and the height of the triangle is h . Substitute 1 2 b h for A 1 in Equation (1). 1 2 b h = 30 b h = 60 Find the area of the square with side 10 cm. A = 10 × 10 = 100 cm 2 Find the remaining area of the region ( A 2 ) : A 2 = A − A 1 (2) Substitute 100 cm 2 for A and 30 cm 2 for A 1 in Equation (2). A 2 = 100 − 30 = 70 cm 2 The value of centroid from the right side of the square is 4 cm. Therefore, the value of centroid from the left side of the square ( x ¯ ) is 6 cm. Draw the square with the dimensions as shown in Figure 1. Refer Figure 1. The equation of the line is y = f ( x ) = h b x + ( 10 − h ) . The expression to find centroid of the remaining region ( x ¯ ) is shown below: x ¯ = 1 A 2 ∫ a b x f ( x ) d x (3) Here, the function is y , the lower limit is a , and the upper limit is b . Substitute 6 cm for x ¯ , 0 for a , and 10 cm for b in Equation (3). 6 = 1 A 2 ∫ 0 10 x f ( x ) d x (4) Split the limits of x -axis into 0 to b and b to 10 in Equation (4). ∫ 0 b x f ( x ) d x + ∫ b 10 x f ( x ) d x = 6 ( A 2 ) (5) Substitute h b x + ( 10 − h ) for f ( x ) in the limit 0 to b , 10 for f ( x ) in the limit b to 10, and 70 cm 2 for A 2 in Equation (5). ∫ 0 b x ( h b x + ( 10 − h ) ) d x + ∫ b 10 x ( 10 ) d x = 6 ( 70 ) ∫ 0 b ( h b x 2 + 10 x − h x ) d x + ∫ b 10 10 x d x = 420 [ h b x 3 3 + 10 x 2 2 − h x 2 2 ] 0 b + [ 10 x 2 2 ] b 10 = 420 [ h 3 b x 3 + 5 x 2 − h 2 x 2 ] 0 b + [ 5 x 2 ] b 10 = 420 (6) Solve Equation (6). ( h 3 b b 3 + 5 ( b ) 2 − h 2 ( b ) 2 ) − ( 0 ) + ( 5 ( 10 ) 2 − 5 b 2 ) = 420 h b 2 3 + 5 b 2 − h b 2 2 + 500 − 5 b 2 = 420 h b 2 3 − h b 2 2 = − 80 h b 2 6 = 80 ( h b ) b = 480 (7) Substitute 60 for bh in Equation (7)

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8, Problem 10P

To determine

The centroid distance from the bottom of the square.

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Single Variable Calculus

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