MATH 230 TEST COMPLETE SOLUTIONS WITH 100% CORRECT ANSWERS

MATH 230 TEST COMPLETE SOLUTIONS WITH 100% CORRECT ANSWERS Complete the front page!!!! [2 points!] Instructions: Show ALL of your work on the pages provided! Explain your reasoning using complete sentences and corr ...

MATH 230 TEST COMPLETE SOLUTIONS WITH 100% CORRECT ANSWERS Complete the front page!!!! [2 points!] Instructions: Show ALL of your work on the pages provided! Explain your reasoning using complete sentences and correct grammar, spelling, a√nd punctuation. All numerical answers MUST be exact; e.g., you should write π instead of 3.14..., 2 instead of 1.414..., and 1 instead of 0.3333... Make sure that your final answer is clearly indicated. This test has eleven pages. Check to make sure you have pages 1 - 11. Calculators, course notes, textbooks, etc. are NOT allowed. Do not write your name on this exam! Write your NetID and your Student ID Number in the spaces provided at the top of this page. Mark your section!! Question 1. Let S be the sphere given by the equation x2 − 8x + y2 + 24y + z2 − 12z = −171 (a) (5 points) Find the center and radius of the sphere S. (b) (5 points) The sphere S intersects the plane z = 2. Find the area enclosed by this intersection. Question 2. Bender and the Robot Santa need to travel from the Planet Express office, which is located at the point (0, 3), to Elzar’s Fine Cuisine, which is located at the point (2, 7). Bender plans to travel along the curve C1 which is parametrized by r1(t) = (5t, 25t2 + 3) Robot Santa plans to travel along a different curve, C2, which is parametrized by r2(t) = (2t, 8t3 − 4t + 3) In both of these parametrizations, t represents the number of hours since the beginning of the journey, and distance is measured in miles. (a) (3 points) Who will reach the destination first, Bender or Robot Santa? (b) (5 points) If they leave the office at the same time, will Bender and Robot Santa crash into each other after they leave the office? (c) (5 points)Set up (but do NOT evaluate) two different integrals which give the lengths of the paths traveled by each robot from the office to the restaurant. Question 3 (8 points). A tow truck pulled a broken-down car horizontally 10 m. The chain attaching the car to the truck is 60◦ above the horizontal. In moving the car, the tow truck performs 100 Joules of work. Supposing the truck applied a constant force along the chain to perform this work, what was the magnitude of that force (in Newtons)? Question 4 (8 points). After a daring leap from a tree branch while chasing a squirrel, Marcel the cat is left dangling from a clothes line hanging between a house and the tree. The segment of rope between Marcel and the tree makes an angle of 45◦ with the horizontal. The segment of rope between Marcel and the house makes an angle of 30◦ with the horizontal. If Marcel weighs 15 lbs, find the tension vectors T1 and T2 in each piece of rope and their magnitudes. Question 5 (2 points each). Match each of the following polar equations with the curve it describes. Write your answer in the space provided: (i) r = 2 cos(θ) + 1 (ii) r = 3 cos(θ) − 2 sin(θ) (iii) r = 3 sin(4θ) (iv) r = 2 sin . θ Σ A. B. C. D. Question 6 (12 points). Let C be the curve parametrized by r(t) = (2 cos t, 2 sin t, et), 0 ≤ t ≤ π. Find the point on C where the tangent line to C is parallel to the plane √3x + y = 1. Question 7. Consider the surface S in R3 defined by y2 = 4 + 12x2 + 12z2. (a) (3 points each) Write the equations for the traces in the planes x = 0, z = 0, y = 2 and y = 4. Provide a sketch for each trace below. Label the points of intersection on the coordinate axes. Question 7(b). (4 points) Determine which of the following is the surface defined by the equation y2 = 4 + 12x2 + 12z2. Circle your answer! Question 8. Let L1 be the line parametrized by r(t) = (1, 2, 3) + t(1, 1, 1) and let L2 be the line give by the equations: x(t) = 7t y(t) = 3t z(t) = 4 (a) (6 points) Determine if the lines L1 and L2 are parallel, intersecting or skew. (b) (9 points) Find an equation for the plane that contains L1 and is parallel to L2. Question 9 (8 points). Suppose a triangle in R3 h√as vertices (0, 0, 0), (1, 1, 1), and (a, a, 1), where a ƒ= 1 is some real number. If the triangle has area 5 2, find all the possible values for a.