Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of θ is L ( θ ) = 4 csc θ + 3 sec θ . a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation 3 sec θ tan θ − 4 csc θ cot θ = 0 0 ∘ < θ < 90 ∘ Solve this equation for θ . b. What is the length of the longest ladder that can be carried around the corner? c. Graph L = L ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that minimizes the length L . d. Compare the result with the one found in part (a). Explain why the two answers are the same.
Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of θ is L ( θ ) = 4 csc θ + 3 sec θ . a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation 3 sec θ tan θ − 4 csc θ cot θ = 0 0 ∘ < θ < 90 ∘ Solve this equation for θ . b. What is the length of the longest ladder that can be carried around the corner? c. Graph L = L ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that minimizes the length L . d. Compare the result with the one found in part (a). Explain why the two answers are the same.
Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length
of the ladder as a function of
is
.
a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation
Solve this equation for
.
b. What is the length of the longest ladder that can be carried around the corner?
c. Graph
,
, and find the angle
that minimizes the length
.
d. Compare the result with the one found in part (a). Explain why the two answers are the same.
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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