Expert: Alon Mandes - 3/7/2009

Question
I have tried both these problems several different ways and have yet to come up with a correct answer.

A spherical balloon is being inflated uniformly.
A. How fast is its radius increasing when the volume is increasing at the rate of 1 [(cc)/(min)] and r = 5 cm?
B. How fast is its surface area changing at that instant?
Submit your answer as a decimal number accurate to within one percent. You may use a calculator for this problem.


A. How fast is the radius changing (in [(cm)/(min)])?
B. How fast is the surface area changing (in [(cm2)/(min)])



A cube of ice is melting uniformly at the rate of 2 [(cc)/(min)].
A. How fast is the edge length changing when the volume of the ice cube is 10 cc?
B. How fast is the surface area changing (in [(cm2)/(min)])?
Submit your answer as a decimal number accurate to within one percent. You may use a calculator for this problem.
A. How fast is the edge length changing (in cm/min)?


B. How fast is the surface area changing (in [(cm2)/(min)])?

Answer
# A spherical balloon is being inflated uniformly.
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(A) V=(4/3)πr³ . It's given that V'(t)=1, thus
   V'=4πr²r'
   1=4π*25*r'
   r'=1/100π
(B) S=4πr²
   S'=8πrr'
   S'=8*π*5*(1/100π)
   S'=2/5 .


# A cube of ice is melting uniformly at the rate of 2 [(cc)/(min)].
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(A) V'=-2 . Edge length is x.
   We know that V=x³. When the volume is 10 then x=10^(1/3) .
   V'(t)=3x²(t)x'(t)
   -2=3[10^(1/3)]²x'(t)
   x'(t)=-0.43
(B) x'(t)=-0.43 -> x(t)=C-0.43t
   S=6x²
   S(t)=6(C-0.43t)²
   S'(t)=12(C-0.43t)*(-0.43)
   S'(t)=-5.16(C-.43t)
Where C is the initial value of the length x.