Introduction to Engineering Mathematics

E154, Fall 2003

Midterm10/29/2003

Directions: for full credit, answer all 4 questions. Bonus question will receive no partial credit. Show all work Problem 1 [25 pts] Let , , and

a) [5 pts] Find the angle between and b) [5 pts] Find the volume of a parallelepiped formed by , , and c) [5 pts] Find the equation of a plane parallel to and and passing through the

point d) [5 pts] Determine a set of parametric equations for the line pointing in the

direction of and passing through the point e) [5 pts] Find the component of in the direction perpendicular to both and

Problem 2 [25 pts] The acceleration of a particle moving in space is given by: . At the velocity of the particle is and the

position of the particle is .

a) [4 pts] Compute the velocity vector as a function of timeb) [4 pts] Compute the position vector as a function of timec) [4 pts] Find the speed of the particle at secd) [4 pts] Find the curvature of the trajectory at sece) [4 pts] Find the normal component of the acceleration vector at secf) [5 pts] To estimate the distance traveled by the particle, one could integrate the

speed of the particle over time. Numerically, this can be implemented using the following two equations:

Assuming , , and sec, write a MATLAB script to determine and display the time it takes for the particle to travel a distance of 10 units. Be sure to properly initialize your variables and use proper syntax as if you were writing a real code. [Hint: use the expression for the speed found in part c). You may want to use a while loop conditioned to iterate for as long as

. Do not forget to increment n]

Problem 3 [25 pts] The volume of a fuel tank consisting of two hemispherical caps of

radius r and a cylindrical section of length h is given by .

The nominal values of the radius and the length are and , respectively.

a) [5 pts] It is known that as the temperature of the tank increases, both the radius and the length increase by the following maximum amounts: and

. Determine the approximate change in the volume of the tank. Express your answer as a multiple of

b) [5 pts] It was experimentally determined that the tank temperature varies sinusoidally in time, such that and . Determine the

rate with which the volume of the tank changes as a function of time

c) [15 pts] A square meter of a hemispherical section costs $100 and a square meter of a cylindrical section costs $50. Use the method of Lagrange multipliers

to find the optimum dimensions of the tank whose volume is such as to

minimize the total cost

Problem 4 [25 pts] Let

a) [5 pts] Find the directional derivative of at in the direction of

b) [5 pts] Find a unit vector in the direction in which increases the most at

c) [5 pts] If the independent variables are constrained such that ,

compute and evaluate it at

d) [5 pts] Determine the equation of a tangent plane to a surface at

e) [5 pts] Suppose you were asked to use MATLAB to make a 3D plot of the level surface in part d) over the range and using an increment of 0.01 in both directions. Write a MATLAB script to execute this task. Be sure to use proper syntax as if you were writing a real code. [Hint: you may need to solve for z first]

BONUS [10 pts] An airplane is entering a vertical loop with the speed of such that its normal acceleration remains constant at 4g. Recall that the curvature can be

expressed as follows: , where is the angle that the

velocity vector makes with respect to the horizontal. Assume that thrust is equal to drag throughout the flight.

vL

mg

a) Apply the second Newton’s law in the tangential direction and show that:

b) Show that the normal acceleration can be expressed as follows:

c) Divide the two equations by each other to obtain a differential equation relating and . Show that:

d) Assuming that when , integrate both sides of the equation in c) and evaluate the constant of integration. Show that the speed of the airplane and the radius of curvature of the path as the plane traverses the loop are given by the following expressions, respectively: