If a snowball melts so its surface area decreases at a rate of 1 cm^2
/min , find the rate at which the diameter
decreases when the diameter is 6 cm?

Question

Ezra Adams · Accepted Answer

#d'(t_0)=-1/(12π) # #"cm/min"# For #color(purple)(t=t_0)# , #A'(t_0)=-1# , #d(t)=6# #<=># #2*r(t)=6# , #d'(t)=2*r'(t)# #<=># #-1=4π*d(t_0)*r'(t_0)# #<=># #<=># #-1=4π*6*r'(t_0)# #<=># #<=># #-1=2π*6*2r'(t_0)# #<=># #<=># #-1=2π*6*d'(t_0)# #<=># #<=># #-1=12π*d'(t_0)# #<=># #<=># #d'(t_0)=-1/(12π)# #"cm/min"#

Charles Arocha · Answer

The diameter is decreasing at a rate of #1/(12pi) \ "cm " "min"^(-1)#

Now let's set up the variables listed below:

{

(D, "Diameter of snowball at time t","(cm)"), (S, "Surface area of snowball at time t", "(cm"^3")"), (t, "time", "(min)") :} #

The standard formula for Surface Area of a sphere with radius #r# is :

# S = 4pir^2 # # \ \ = 4pi(D/2)^2 # # \ \ = piD^2 #

Differentiating wrt #D#, we have:

# (dS)/(dD) = 2piD #

We are informed that the surface area decreases at a constant rate of 1, so:

# (dS)/(dt) = -1 #

And, we seek the value of #(dD)/dt# when #D=6#, which we expect to be negative (as the volume is decreasing). Applying the chain rule, we have:

# (dD)/(dt) = (dD)/(dS) * (dS)/(dt) # # \ \ \ \ \ \ = 1/(2piD) * (-1) # # \ \ \ \ \ \ = -1/(2piD) #

So then:

# [ (dD)/(dt) ]_(D=5) = -1/(2pi * 6) = -1/(12pi)#

Thus the diameter is decreasing at a rate of #1/(12pi) \ "cm " "min"^(-1)#

Ezra Adams · Answer

undefined To find the rate at which the diameter decreases when the diameter is 6 cm, use the formula for the surface area of a sphere:

Surface area (A) = 4πr^2

Differentiate both sides with respect to time (t), and use the chain rule:

dA/dt = 8πr(dr/dt)

Given dA/dt = -1 cm^2/min (since the surface area decreases), and when the diameter is 6 cm, the radius (r) is half of the diameter, so r = 3 cm.

Plug in the values and solve for dr/dt:

-1 = 8π(3)(dr/dt)
dr/dt = -1 / (8π(3))

Approximately dr/dt ≈ -0.042 cm/min.

So, the rate at which the diameter decreases when the diameter is 6 cm is approximately -0.042 cm/min.

If a snowball melts so its surface area decreases at a rate of 1 cm^2 /min , find the rate at which the diameter decreases when the diameter is 6 cm?

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