Calculus AP

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Can someone help me step by step how to find the answer to this problem.

The amount of air in a balloon at time t is given by the equation
a(t)=3tsint+t^2 where t is measured in seconds and the amount of air is measured in cubic centimeters.

a. what is the amount of air in the balloon after 10 seconds.
b. is the amount of air in the balloon increasing or decreasing at t=4 seconds and at what rate ?
c. When does the amount of air in the balloon reach 30 cubic centimeters?


  • Calculus AP -

    a. t=10
    Evaluate a(t)=3tsint+t^2 using t=10, i.e.

    find a'(t)=da/dt using the product rule and the sum rule. Then evaluate a'(4) to see if it is positive (increasing) or negative (decreasing).

    Product rule : d(uv)/dt = udv/dt+vdu/dt
    Sum rule : d(p+q)/dt = dp/dt + dq/dt

    In the given expression for a(t),
    a'(t) = d(3tsint+t^2)
    =d(3t.sin(t) )/dt + d(t²)/dt
    =3sin(t)+3t.cos(t) + 2t
    Evaluate a'(4) to verify the sign.

    c. When it reaches 30 cm³,
    3tsint+t^2 = 30
    Solve for t by trial and error or newton's method. The answer should be around 6.

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