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SPRING 2013 – Calculus 101C – Test 1 – Chapter 11 Show Work for Credit 1 Tell whether the two lines intersect, are parallel, are skew, or coincide. If they intersect, give the point of intersection. x +1 y − 3 z − 2
=
=
2
−1
1
and x +1 y +1 z − 3
=
=
2
3
−4
2 Suppose that the wind is blowing with a 500 lb magnitude force in the direction of S45W behind a boat’s sail. How much work does the wind perform in moving the boat south a distance of 60 ft? 3 Find the area of the triangle whose vertices are ( 4 , 3 , 1 ) , ( 2 , 2 , 1 ) , and ( 1 , 1 , 0 ). 4 Find the torque and the magnitude of the torque when a force F = 3i -­‐ 2j + k is applied to a lever at the point ( 3 1 2 ) if the lever goes from the point ( 1 0 3 ) to the point ( 3 1 2). 5a Find the vector with initial point A and terminal point B. 𝐴 = 3 , 4 𝐵 = 2 3 , 5 5b Find a unit vector in the same direction as the vector in 5a. 5c Write the vector in 5a in polar form. 6 The thrust of an airplane’s engine produces a speed of 900 mph in still air. The plane is aimed in the direction 6 3 6 and the wind velocity is 0 200 −100 mph. Find the velocity vector of the plane with respect to the ground AND find the speed. Reduce your answer for speed as much as possible but leave the radical sign. (meaning: 50 = 5 2 ) 7 Find the equation of the line that goes through the point 1 , 1 , 1 and is orthogonal to the two vectors 4 0 1 and 0 1 −2 . Write the equation in symmetric form. 8 A curve in a railroad track has the shape of the parabola y
= x 2 / 4. If a train is loaded so that its scalar normal component of acceleration cannot exceed 25 units/sec2 , what is its maximum possible speed as it rounds the curve at ( 2 3 , 3 )? 9 A baseball is hit 2 feet above ground level at an angle of 30 degrees with a speed of 80 ft/sec. What is the maximum height reached by the ball? First, derive the position vector function from the acceleration vector function then answer the question. 10a Derive the formula for centripetal force from a vector function that represents a circle of radius R with angular velocity ω . b A man holds onto a pail of water weighing six pounds and swings it in a vertical circle with radius of four feet. What is the smallest value of ω required to keep the water from spilling from the pail? 11 Suppose a road is a level curve that is a circle of radius 300 feet, and the average speed on this curve is 60 mi/hr. Also assume the vehicle weighs 3,000 lbs. (5280 ft = 1 mi) Find the frictional force between the tires and the road so that the car stays on the circular path and does not skid. 12 Find the tangent LINE to 𝑟 𝑡 = ln 𝑡
𝑡!
5𝑡 at the point ( 0 , 1 , 5 ) 13 Find the arc length parameterization of the line segment from 1 , 2 to −5 , 10 . 14 A projectile of mass 1 slug is launched from 50 feet above ground level toward the West at 10 3 feet / second, at an angle of 60 degrees from the horizontal. If the spinning of the projectile applies a steady southerly Magnus force of 8 pounds to the projectile. (we have an acceleration due to the Magnus force of 𝑎! =
8 0 0 that must be added to the acceleration vector that is due to gravity) Find the vector function that describes the path of the projectile and sketch the trajectory of the projectile on the coordinate system below: z North West y East x South 15 Find the point (or points) where the ellipse 4 x 2 + 25 y 2 = 100 has maximum curvature. Parameterize the curve, find the curvature function, and find the points that maximize the curvature function. Use calculus – ie: Derivative and First Derivative Test Draw a picture of the ellipse to verify your computation.