Respuesta :
Answer:
dS/dt = 0.33 cm²/min
Explanation:
Volume of sphere is given by;
V = (4/3)πr³
We are told that gas is escaping from a spherical balloon 4 cm³ per minute. Thus; dV/dt = 4 cm³/min
Now, let's find the rate at which the radius is changing.
dr/dt = (dV/dt) ÷ (dV/dr)
Now, dV/dr = 4πr²
Thus;
dr/dt = 4/(4πr²)
dr/dt = 1/(πr²)
Now, surface area of sphere is given by;
S = 4πr²
Thus, dS/dr = 8πr
Now, let's find the rate at t which area is shrinking.
dS/dt = (dS/dr) × (dr/dt)
Plugging relevant terms, we have;
dS/dt = 8πr × 1/(πr²)
dS/dt = 8/r
At r = 24 cm, we have;
dS/dt = 8/24
dS/dt = 1/3
dS/dt = 0.33 cm²/min
