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Q. Write down 30 uses of Surface Integration in computer science Engineering
Find the volume in the first octant bounded by x+y+z=9, and the inside cylinder 3y=27-x^3
Find the relative extrema of π π₯ = 3π₯^5 β 5π₯^2.
What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoidΒ x2/9+y2/16+z2/36=1? Show that the answer you get gives you the largest volume.(DO NOT USE LAGRANGE MULTIPLIERS)
a. When we cough, the trachea (windpipe) contracts to increase the velocity
of the air going out. This raises the questions of how much it should contract
to maximize the velocity and whether it really contracts that much when we
cough.
Under reasonable assumptions about the elasticity of the tracheal wall and
about how the air near the wall is slowed by friction, the average flow velocity
y can be modeled by the equation π¦ = π( π , ,
0 β π)π
2
ππ/π ππ
π0
2 < π < π0
where π is the rest radius of the trachea in centimeters and is a positive
π
constant whose value depends in part on the length of the trachea. Show that
π¦ is greatest when π = (2/3)π that is, when the trachea is about
33%
contracted. The remarkable fact is that π βray photographs confirm that the
trachea contracts about this much during a cough.
b. Take π to be and to be and graph over the interval .
0. 5 π 1 π¦ 0 < π < 0. 5
Compare what you see with the claim that π¦ is at a maximum when
π = (2/3)π .
Another representation, in polar coordinates, of the point (1,5pi/6) is (___, 11pi/6)
Solve the equation π = β«10π2t ππ‘ for T range 0 to T.
What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoidΒ x2/9+y2/16+z2/36=1? Show that the answer you get gives you the largest volume.(DO NOT USE LAGRANGE MULTIPLIERS)
A soft drink manufacturer requires services of your software house to automate the process of can designing. The available material for the bottom and top of the can cost 20cents/cm^2. and for the sides the material cost is 10cents/cm^2. The company is considering two design options; one design does not include the lid and is an open top design, and the other design includes the lid and thus is a closed top design. The company wants your software house to provide them with a MATLAB code which will take desired volume, and design (open top or closed top) as input and give dimensions and surface area of the can that will result in the minimum cost as output for the specified volume and design. To test your code the company has asked you to use 275cm^3 as volume for both designs. Furthermore, if the budget of the company is $2 million, how many cans of the required design will the company be able to make?
A boat is pulled into a dock by means of a rope attached to a pulley on the dock, Figure 2.1. The rope is attached to the bow of the boat at a point 1 m below the pulley. If the rope is pulled through the pulley at a rate of 1 m/sec, at what rate will the boat be approaching the dock when 10 m of rope is out
