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Math Help - Word Problem involving Integration

  1. #1
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    Word Problem involving Integration

    Any help as to how to get started or get through the problem will be much appreciated:

    An oil spill is spreading in a circular pattern from a damaged tanker on a calm sea. The radius of the circle is growing at a rate of r'(t)=40 times the radical of t feet per minute.

    A)How much does the radius grow from time t= 100 to t= 225?
    B) When will the radius be 1,000 feet long?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by blurain View Post
    Any help as to how to get started or get through the problem will be much appreciated:

    An oil spill is spreading in a circular pattern from a damaged tanker on a calm sea. The radius of the circle is growing at a rate of r'(t)=40 times the radical of t feet per minute.

    A)How much does the radius grow from time t= 100 to t= 225?
    B) When will the radius be 1,000 feet long?
    r(t) = \int r'(t)~dt = \int 40 \sqrt{t}~dt = 40 \int t^{\frac 12}~dt ----> use the power rule

    (A) we want \int_{100}^{225} r'(t)~dt = r(225) - r(100)

    or you can use the second fundamental theorem of calculus if you which, but direct integration is more fun

    (B) we want r(t) = 1000 and solve for t
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  3. #3
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    Is r(t)=80/3 t^3/2 the correct function for the derivative r'(t)= 40 times the radical of t?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by blurain View Post
    Is r(t)=80/3 t^3/2 the correct function for the derivative r'(t)= 40 times the radical of t?
    yes, but it should really be \frac {80}3t^{3/2} + C but we know that C = 0, so it's fine
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