The Subaru telescope is a large optical-infrared telescope at the summit of Mauna Kea, Hawaii. The telescope has a parabolic mirror 8.2 m in diameter with a focal length of 15 m . a. Suppose that a cross section of the mirror is taken through the vertex, and that a coordinate system is set up with 0 , 0 placed at the vertex. If the focus is 0 , 15 , find an equation representing the curve. b. Determine the vertical displacement of the mirror relative to horizontal at the edge of the mirror. That is, find the y value at a point 4.1 m to the left or right of the vertex. c. What is the average slope between the vertex of the parabola and the point on the curve at the right edge?
The Subaru telescope is a large optical-infrared telescope at the summit of Mauna Kea, Hawaii. The telescope has a parabolic mirror 8.2 m in diameter with a focal length of 15 m . a. Suppose that a cross section of the mirror is taken through the vertex, and that a coordinate system is set up with 0 , 0 placed at the vertex. If the focus is 0 , 15 , find an equation representing the curve. b. Determine the vertical displacement of the mirror relative to horizontal at the edge of the mirror. That is, find the y value at a point 4.1 m to the left or right of the vertex. c. What is the average slope between the vertex of the parabola and the point on the curve at the right edge?
The Subaru telescope is a large optical-infrared telescope at the summit of Mauna Kea, Hawaii. The telescope has a parabolic mirror
8.2
m
in diameter with a focal length of
15
m
.
a. Suppose that a cross section of the mirror is taken through the vertex, and that a coordinate system is set up with
0
,
0
placed at the vertex. If the focus is
0
,
15
,
find an equation representing the curve.
b. Determine the vertical displacement of the mirror relative to horizontal at the edge of the mirror. That is, find the
y
value at a point
4.1
m
to the left or right of the vertex.
c. What is the average slope between the vertex of the parabola and the point on the curve at the right edge?
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
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?
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
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