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solCal3samplemid1

# solCal3samplemid1 - SOLUTIONS TO CALCULUS III SAMPLE...

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SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 1 Problem 1. (20 points) (a)Find an equation of the largest sphere with center (5 , 4 , 9) that is contained in the first octant. (b)Find an equation for the surface consisting of all points P for which the distance from P to the x -axis is twice the distance from P to the yz -plane. Identify the surface. Solution: (a) Imagine that a sphere is growing up from the neighborhood of that point. Which coordinate plane is the first one that is contacted by this growing sphere? It is the nearest coordinate plane, in this case, the xz -plane. At this time the radius of the sphere is 4. So, an equation of this largest admissible sphere is ( x 5) 2 + ( y 4) 2 + ( z 9) 2 = 16 . (b) Let the coordinates of P is ( x, y, z ), then the distance from P to x -axis is radicalbig y 2 + z 2 . The distance from P to yz -plane is | x | . So the equation for the set of all points satisfying the condition is radicalbig y 2 + z 2 = 2 | x | . Square both sides and write in the standard form, y 2 + z 2 = 4 x 2 . We identify this surface as a cone . Problem 2. (20 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: 1

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2 SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 1 0.8 x 9.8 N 8 cm 4 m 4m x y θ θ θ θ T T The angle θ is determined by tan θ = 8 cm 4 m = 0 . 08 4 = 0 . 02 .
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solCal3samplemid1 - SOLUTIONS TO CALCULUS III SAMPLE...

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