In this page question4 in application problems we are going to see solution of first question
The radius of a spherical balloon is increasing at the rate of 4 cm/sec. Find the rate of increases of the volume and surface area when the radius is 10 cm.
Let "V' be volume of spherical balloon and "S" be the surface area.
Here we need to find "dV/dt" and "dS/dt"
dr/dt = 4 cm/sec and r = 10 cm
Volume of the spherical balloon (V) = (4/3) Π r³
Differentiate with respect to t
dV/dt = (4/3) Π 3 r² (dr/dt)
dV/dt = (4/3) Π 3 (10)² (4)
dV/dt = 16 Π (10)²
dV/dt = 16 Π (100)
dV/dt = 1600 Π cm³/sec
Surface area of the spherical balloon S = 4 Π r²
differentiate with respect to t
dS/dt = 4 Π 2r (dr/dt)
dS/dt = 8 Π r (dr/dt)
dS/dt = 8 Π (10) (4)
dS/dt = 80 Π (4)
dS/dt = 320 Π cm²/sec
(1) The radius of a circular plate is increasing in length at 0.01 cm per second. What is the rate at which the area is increasing when the radius is 13 cm?
(2) A square plate is expanding uniformly each side is increasing at the constant rate of 1.5 cm/min. Find the rate at which the area is increasing when the side is 9 cm.
(3) A stone thrown into still water causes a series of concentric ripples. If the radius of outer ripple is increasing at the rate of 5 cm/sec,how fast is the area of the distributed water increasing when the outer most ripple has the radius of 12 cm/sec.
(5) A balloon which remains spherical is being inflated be pumping in 90 cm³/sec. Find the rate at which the surface area of the balloon is increasing when the radius is 20 cm.