Calculus AP

posted by .

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?


THANK YOU!

  • Calculus AP -

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

    b.
    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.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    Find the equation of the tangent to the curve y=2-e^-x at the point where x=1. Can someone please do this step by step and explain?
  2. Calculus

    t: 0 2 5 7 11 12 r'(t): 5.7 4.0 2.0 1.2 0.6 0.5 The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon in feet is modeled by a twice-differentiable function of r of time …
  3. Algebra

    in that equation I have to solve for x. 5. x^3 + 2x^2 - 9x = 18 in this word problem i have to solve my factoring. 8. Suppose the altitude of a rising hot-air balloon is given by h = 0.04 t2 + 2t, where “t” is the time in seconds …
  4. Physics

    Hello recently I was asked to find the time it takes a satalite that is geocentric (that always stays above the same exact spot of the earth) so... I was given the radius of 61 earth radius or simply 3.89 E 6 m mass of earth what G …
  5. Calculus

    The rate of change R, in kilometers per hour, of the altitude of a hot air balloon is given by R(t)=t^3-4t^2+6 for time on the interval [0,4], where t is measured in hours. Assume the balloon is on ground level. What is the maximum …
  6. Calculus

    Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. f(x) = 16 - 10 x Step 2: f(x + h) - f(x) = Step 4: f'(x) = lim(h->0)(f(x + h)- f(x))/h =
  7. ALGEBRA HELP

    When solving the equation, what property was used to go from step 3 to step 4?
  8. Help me! Geometry

    A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm^3 of air. How much air does the balloon hold when the design is 10 cm wide?
  9. calculus help me plz too hard show step

    charge Q coulombs at time t seconds is given by the differential equation RdQ/dt+Q/C=0, where c is the capacitance in farafd and R the resistance in ohms.solve the equation for Q given that Q=Qo where t=0 plz show step plz plz plz
  10. Math

    I need help solving the problem not the answer. (Multiple Choice) The steps below show the incomplete solution to find the value of x for the equation 7x + 8 − 3x = −6 + 10: Step 1: 7x + 8 − 3x = −6 + 10 Step 2: 7x + 8 − …

More Similar Questions