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 ±rst 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 ±rst 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 r 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 r 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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