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solC3-1sample08

# solC3-1sample08 - SOLUTIONS TO CALCULUS III SAMPLE MIDTERM...

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SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 1 Problem 1. (10 points) A clothesline is ted between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the clothesline. Solution: 0.8 x 9.8 N b 8 cm 4 m 4 m x y θ θθ θ T 2 T 1 The angle θ is determined by tan θ = 8 cm 4 m = 0 . 08 4 = 0 . 02 . Denote the tensions by v T 1 and v T 2 . In the coordinate system, the Newton’s ±rst law gives two equations, in x -component and y -component respectively. −| v T 1 | cos θ + | v T 2 | cos θ = 0 | v T 1 | sin θ + | v T 2 | sin θ = 0 . 8 × 9 . 8 (N) . The ±rst equation says | v T 1 | = | v T 2 | , and then the second becomes 2 | v T 1 | sin θ = 0 . 8 × 9 . 8 (N). Compute sin θ , sin θ = 1 csc θ = 1 1 + cot 2 θ = 1 r 1 + ( 1 0 . 02 ) 2 = 1 1 + 2500 1 2500 = 0 . 02 The magnitudes of the tensions are | v T 1 | = | v T 2 | = 0 . 8 × 9 . 8 2 sin θ = 0 . 8 × 9 . 8 0 . 04 = 20 × 9 . 8 = 196 N 1

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2 Problem 2. (10 points) Given two planes z = x + y, 2 x 5 y z = 1 .
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solC3-1sample08 - SOLUTIONS TO CALCULUS III SAMPLE MIDTERM...

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