Calculus Answers

A container with square base, vertical sides, and open top is to have a volume of 1000 . Find the dimensions of the container with minimum surface area.

A church window consisting of a rectangle topped by a semicircle is to have a perimeter p. Find the radius of the semicircle if the area of the window is to be a maximum.

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 𝑓 𝑥 = 3𝑥^5 − 5𝑥^2.

What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoid 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 𝑦 = 𝑐( 𝑟 , ,

0 − 𝑟)𝑟

2

𝑐𝑚/𝑠𝑒𝑐

𝑟0

2 < 𝑟 < 𝑟0

where 𝑟 is the rest radius of the trachea in centimeters and is a positive

𝑐

constant whose value depends in part on the length of the trachea. Show that

𝑦 is greatest when 𝑟 = (2/3)𝑟 that is, when the trachea is about

33%

contracted. The remarkable fact is that 𝑋 −ray photographs confirm that the

trachea contracts about this much during a cough.

b. Take 𝑟 to be and to be and graph over the interval .

0. 5 𝑐 1 𝑦 0 < 𝑟 < 0. 5

Compare what you see with the claim that 𝑦 is at a maximum when

𝑟 = (2/3)𝑟 .

Another representation, in polar coordinates, of the point (1,5pi/6) is (___, 11pi/6)

Solve the equation 𝑆 = ∫10𝑒^2t 𝑑𝑡 for T range 0 to T.

What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoid x²/9+y²/16+z²/36=1? Show that the answer you get gives you the largest volume.(DO NOT USE LAGRANGE MULTIPLIERS)