MA241 001
Test 2B (100 pts)
Miss Bosko
• Write your name and seat code on the front of your Blue Book.
• Start each new problem at the top of a page (either front or back).
• Only work written in the Blue Book will be graded.
• Show all work. Answers without work or with ambiguous work will receive a zero.
• The only permissible electronic device is a scientific calculator.
(1) (10pts) Find the exact average value of f (x) = sin 3x on the interval [−π/3, π/3].
(2) (10pts) For what value(s) of r does the function y = ert satisfy y !! + 4y ! − 12y = 0?
(3) (10pts) Find all equilibrium solutions to the differential equation y ! = x cos y.
(4) A tank contains 100 L of milk. A mixture that contains 0.05 kg of cocoa per liter
of milk enters the tank at a rate of 5 L/min. Another mixture that contains 0.03
kg of cocoa per liter of milk enters the tank at a rate of 8 L/min. The chocolate
milk is mixed thoroughly and exits the tank at a rate of 4L/min. Let y(t) be the
amount of cocoa in the tank at time, t.
(a) (10pts) Determine the differential equation dy
dt that models this problem. Simplify, but do not solve.
(b) (8pts) Solve the differential equation.
(c) (4pts) What happens to the amount of cocoa in the tank in the long run?
(5) (20pts) Water is found on the moon where acceleration due to gravity is 1.6 m/s2 .
Suppose a hemisphere-shaped crater with radius r = 8m is filled with water (see
picture). Determine the work needed to pump all the water out of the crater.
(6) D’Angleo is feeding Wee-bay’s fish while he is away on business. The aquarium is
6 ft long, 2 ft wide, and 3 ft deep and filled with water.
(a) (6pts) Find the hydrostatic force on the bottom of the tank.
(b) (10pts) Find the hydrostatic force on the side with dimensions 2 ft by 3 ft.
(7) True or False? (3 pts each)
(a)
Using Euler’s method with a step size of 0.5 to find the solution
of the initial-value problem y ! = y − 2x, y(1) = 0 yields y1 = −1.
(b)
A function with average value 0 between x = 2 and x = 3 will
obtain that average value exactly once.
(c)
Hooke’s law states that the force required to maintain a spring
stretched x units beyond its natural length is proportional to x2 .
(d)
An orthogonal trajectory to a family of circles centered at the
origin is the line y = x.