related rates problem-calculus

posted by .

A spherical balloon is being filled with air in such a way that its radius is increasing at a rate of 2 centimeters per second. At what rate is the volume of the balloon changing at the instant when its surface has an area of 4 pi square centimeters?

  • related rates problem-calculus -

    The balloon is spherical, so
    V = (4/3)(pi)r^3
    SA = 4(pi)r^2

    Differentiate both with respect to time.

    dV/dt = 4(pi)r^2(dr/dt)
    dSA/dt = 8(pi)r(dr/dt)

    We're given the rate of change in SA and need to find the rate of change in volume. Let's write an equation for it.

    (dV/dt)/r = 4(pi)r(dr/dt)
    (dSA/dt)/2 = 4(pi)r(dr/dt)

    (dV/dt)/r = (dSA/dt)/2
    (dV/dt) = (r/2)(dSA/dt)

  • related rates problem-calculus -

    Edit: actually we're given dr/dt and SA.

    Since dV/dt = 4(pi)r^2(dr/dt) and SA = 4(pi)r^2, dV/dt = SA(dr/dt).

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calc

    hydrogen is being pumped into a spherical balloon at the rate of 250 cubic inches per minute. in a and b, you must provide an equation with their proper numerical factors when showing the relationships inquired A) write an equation …
  2. Calculus

    Air is being pumped into a spherical balloon so that its volume increases at a rate of 30 cubic centimeters per second. How fast is the surface area of the balloon increasing when its radius is 19 cm?
  3. calculus

    Air is being pumped into a spherical balloon so that its volume increases at a rate of 80 cubic centimeters per second. How fast is the surface area of the balloon increasing when its radius is 8 cm?
  4. AP calculus AB

    A spherical balloon is being filled with air in such a way that its radius is increasing at a rate of 2 centimeters per second. At what rate is the volume of the balloon changing at the instant when its surface has an area of 4 pi …
  5. related rates calculus problem

    A balloon is being filled with air at a rate of 3 cubic meters per minute. Find the rate at which the radius is changing when the volume is 113.0973355 cubic meters.
  6. help on calculus

    A spherical balloon is being filled with air in such a way that its radius is increasing at a rate of 2 centimeters per second. At what rate is the volume of the balloon changing at the instant when its surface has an area of 4 pi …
  7. Word Problem-Calculus

    A spherical balloon is being inflated in such a way that its radius increases at a rate of 3 cm/min. If the volume of the balloon is 0 at time 0, at what rate is the volume increasing after 5 minutes?
  8. math

    Air is pumped into a balloon such that its volume increases at the rate of 75cm^3 per second. It is assumed that the balloon is spherical all the time. Find in term of pi the rate at which the radius of the balloon is increasing when …
  9. Calculus

    A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2cm/min. At what rates are the volume and surface area of the balloon increasing when the radius is 5cm?
  10. Math - Related Rates

    Air is being pumped into a spherical balloon at a rate of 2 cubic meters per minute. At what rate is the radius changing when the volume of the balloon is 36pi cubic meters?

More Similar Questions