A snowball is melting in the sun. It is noted that its surface decreases at a ra

Question

A snowball is melting in the sun. It is noted that its surface decreases at a rate of 4 cm^2/s at the moment when its diameter is 4/pi cm. The goal here is to determine the rate at which the diameter varies at that same moment. To solve this problem, let x be the diameter of the snowball in cm, A its area in cm^2, and t the time in seconds (s). Express A as a function of x. What is the value of da/dx when x = 4/pi ? Give the exact value. da/dx = Using our previous results, give the (exact) value of dx/dt when x = 4/pi cm. Beware of signs, remember that the surface area of the snowball is decreasing with time! dx/dt = cm/s.

Explanation / Answer

a)
A = 4*pi*(x/2)^2
= pi*x^2

b)
dA/dx = pi*2*x
= pi*2*4/pi
= 8 cm

c)
A = pi*x^2
differentiate with respect to t,
dA/dt = pi*2*x*dx/dt
put values,
-4 = pi*2*(4/pi)*dx/dt
-4 =2*4*dx/dt
dx/dt = -0.5 cm/s

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