True or false related rates question?

True or false: If the radius of a circle is expanding at a constant rate, then its circumference is increasing at a constant rate. Justify your answer.True or false related rates question?
This is true.


Proof:


Let r represent the radius.


Let t represent time.


Let let c represent the circumference.


Let 螖 mean change.


Recall that the cirumference of a circle is given by 2*pi*r.





Suppose the radius of a circle is expanding at a constant rate. Thus, 螖r/螖t = k, where k is some constant. 螖r/螖t = (final_r - initial_r)/螖t.


Consider the rate at which the circumference is changing, given by 螖c/螖t = (final_c - initial_c)/螖t (but remember c=2*pi*r)


= (final_2*pi*r - initial_2*pi*r)/螖t = 2*pi*(final_r - initial_r)/螖t = 2*pi*螖r/螖t (But remember 螖r/螖t = k)


= 2*pi*k


Now since 2 is a constant, pi is a constant, and k is a constant, then 2*pi*k is a constant, and we have 螖c/螖t = 2*pi*k. Therefore the circumference is increasing at a constant rate.//





Hope this helps you out!!True or false related rates question?
2*pi*r=c


assume we add some length to r, we will call it x, then 2*pi(r+x)=2*pi*r+2*pi*x, so as you can see for any value for x, the circumference will increase by a rate of 2*pi of that number





so the answer is true, for every unit of length you add to the radius, the circumference will increase by 2*pi





c=2*pi*r


dc/dr=2*pi, as you can see from the calculus aspect of this the change in circumference with respect to the raidus is a constant of 2*pi which verifies the result from above
C = 2蟺(r)


dC/dt = 2蟺(dr/dt)


Since dr/dt is constant, dC/dt is constant.


Yes, it will be increasing at a constant rate.
Yes because 2TTr is the formula for circumference so yes it would be increasing at a constant rate as well but only probably 1 1/2 times faster.
TRUE!

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