A person coughs when a foreign object is in the windpipe. The velocity of the co

Question

A person coughs when a foreign object is in the windpipe. The velocity of the cough depends on the size of the object. Suppose a person has a windpipe with a 16-mm radius. If a foreign object has a radius r, in millimeters, then the velocity V, in millimeters/second, needed to remove the object by a cough is given by the following equation, where k is some positive constant. For what size object is the maximum velocity needed to remove the object?

V(r)=k (16r^2-r^3), 0<r<16

An object that has a radius of _____ will need maximum velocity to remove it.

Explanation / Answer

given V(r)=k (16r2-r3)

V'(r)=k (16*2r2-1- 3r3-1)
V'(r)=k (32r- 3r2)

V''(r)=k (32*1- 3*2r2-1)
V''(r)=k (32- 6r)

for maximum velocity ,V'(r)=0, V"(r)<0

k (32r- 3r2)=0

=>rk (32- 3r)=0

=>32-3r=0

=>r=32/3

V''(32/3)=k (32- 6*(32/3))

V''(32/3)=k (32- 64)

V''(32/3)=-32k

v"(32/3) <0 , since k is positive constant

so , An object that has a radius of __32/3 millimeters___ will need maximum velocity to remove it.

An object that has a radius of __10.67millimeters___ will need maximum velocity to remove it.

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