1.Suppose that a metal rod of length 40 cm is heated to a temperature of 80degrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusioncoefficient a = 1 You will get the following set of equations:ди² uatəx²u(x,0) = 80u(0, t) = 0,=u(40, t) = 0a. Use separation of variables to solve this equation. Show all your work.b. Write the solution as an infinite sum.c. Sketch a graph of the solutions at t=0, t=1, t=10, t=50.d. Which point/points on the metal rod are the hottest as time increases.e. Approximately how long will it take for the entire rod to cool below 5 degreesCelsius? You may use a single term expansion of the series in yourapproximation.f. Describe following limit and find the solution: lim u(x, t).t→∞

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Suppose that a metal rod of length 40 cm is heated to a temperature of egrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion oefficient a = 1 You will get the following set of equations: ди at u(x, 0) u(0, t) = 0, ² u 0x2 = 80 u(40, t) = 0 Use separation of variables to solve this equation. Show all your work. . Write the solution as an infinite sum. Sketch a graph of the solutions at t=0, t=1, t=10, t=50.

Suppose that a metal rod of length 40 cm is heated to a temperature of egrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion oefficient a = 1 You will get the following set of equations: ди at u(x, 0) u(0, t) = 0, ² u 0x2 = 80 u(40, t) = 0 Use separation of variables to solve this equation. Show all your work. . Write the solution as an infinite sum. Sketch a graph of the solutions at t=0, t=1, t=10, t=50.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter2: Functions And Graphs

Section2.6: Proportion And Variation

Problem 17E: Find the constant of proportionality. y is directly proportional to x. If x=30, then y=15.

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Question

1.
Suppose that a metal rod of length 40 cm is heated to a temperature of 80
degrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion
coefficient a = 1 You will get the following set of equations:
ди
² u
at
əx²
u(x,0) = 80
u(0, t) = 0,
=
u(40, t) = 0
a. Use separation of variables to solve this equation. Show all your work.
b. Write the solution as an infinite sum.
c. Sketch a graph of the solutions at t=0, t=1, t=10, t=50.
d. Which point/points on the metal rod are the hottest as time increases.
e. Approximately how long will it take for the entire rod to cool below 5 degrees
Celsius? You may use a single term expansion of the series in your
approximation.
f. Describe following limit and find the solution: lim u(x, t).
t→∞

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