A 120 -ft flight control tower in the shape of a hyperboloid has hyperbolic cross sections perpendicular to the ground. Placing the origin at the bottom and center of the tower, the center of a hyperbolic cross section is 0 , 30 with one focus at 15 10 , 30 and one vertex at 15 , 30 . All units are in feet. a. Write an equation of a hyperbolic cross section through the origin. Assume that there are no restrictions on x or y . b. Determine the diameter of the tower at the base. Round to the nearest foot. c. Determine the diameter of the tower at the top. Round to the nearest foot.
A 120 -ft flight control tower in the shape of a hyperboloid has hyperbolic cross sections perpendicular to the ground. Placing the origin at the bottom and center of the tower, the center of a hyperbolic cross section is 0 , 30 with one focus at 15 10 , 30 and one vertex at 15 , 30 . All units are in feet. a. Write an equation of a hyperbolic cross section through the origin. Assume that there are no restrictions on x or y . b. Determine the diameter of the tower at the base. Round to the nearest foot. c. Determine the diameter of the tower at the top. Round to the nearest foot.
Solution Summary: The author calculates the equation of a hyperbolic cross section through the origin.
A
120
-ft
flight control tower in the shape of a hyperboloid has hyperbolic cross sections perpendicular to the ground. Placing the origin at the bottom and center of the tower, the center of a hyperbolic cross section is
0
,
30
with one focus at
15
10
,
30
and one vertex at
15
,
30
.
All units are in feet.
a. Write an equation of a hyperbolic cross section through the origin. Assume that there are no restrictions on
x
or
y
.
b. Determine the diameter of the tower at the base. Round to the nearest foot.
c. Determine the diameter of the tower at the top. Round to the nearest foot.
2. Consider the following:
Prove that x, x2, and 1/x are the solutions to the homogeneous equation
corresponding to x³y"" + x²y" + 2xy' + 2y = 2x4.
b. use variation of parameters to find a particular solution and complete the general
solution to the differential equation. I am interested in process. You may use a
computer for integration, finding determinants and doing Kramer's.
3. A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot
mechanism that has a damping constant of 0.25 lb-sec./ft. and is acted on by an external
force of 4 cos 2t lb.
a. Set-up the differential equation and initial value problem for the system.
b. Write the function in phase-amplitude form.
C.
Determine the transient solution to the system. Show your work.
d. Determine the steady state of this system. Show your work.
e.
Is the system underdamped, overdamped or critically damped? Explain what this
means for the system.
4. Suppose that you have a circuit with a resistance of 20, inductance of 14 H and a
capacitance of 11 F. An EMF with equation of E(t) = 6 cos 4t supplies a continuous charge
60
to the circuit. Suppose that the q(0)= 8 V and the q'(0)=7. Use this information to answer the
following questions
a. Find the function that models the charge of this circuit.
b. Is the circuit underdamped, overdamped or critically damped?
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Finding The Focus and Directrix of a Parabola - Conic Sections; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=KYgmOTLbuqE;License: Standard YouTube License, CC-BY