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Math(Please check)

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I posted this question a few days ago and someone answered it but I just wanted to make sure that I copied down the correct answers and didn't misunderstand anything.

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Gas is escaping a spherical balloon at the rate of 2ft^3/min. How fast is the surface area shrinking (ds/dt) when the radius is 12ft? ( A sphere of raius r has volume= 4/3 pir^3 and surface area S=4pir^2.)

Remember that ds/dt=ds/dr X dr/dt

Step 1: Find ds/dr
S=4pir^2 so ds/dr= (4pi)(2r) = 8pi r

Step 2: Fid dr/dt (Hint: dv/dt=dv/dr X dr/dt)

V=4/3pir^3 dv/dr=4pir^2 2=(4pir^2)(dr/dt) dr/dt=1/2pir^2

Step 3: Find ds/dt

ds/dt= (8pir)(1/2pir^2) = 4/r

Step 4: Evaluate ds/dt when the radius is 12ft.

r=12 (8)(3.14)(12) X (1/2(3.14)(12)^2) =4/12


Are these correct?

  • Math(Please check) -

    correct, except reduce 4/12 to 1/3

    I might have done it this way

    V = (4/3)πr^3
    dV/dt = 4πr^2 dr/dt , but when r=12, dV/dt = 2
    2 = 4π(144) dr/dt
    dr/dt = 1/(288π)

    A = 4πr^2
    dA/dt = 8πr dr/dt
    = 8π(12)(1/(288π) = 1/3

  • Math(Please check) -

    Ok thank you!

  • Math(Please check) -


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