Final answer:
The rate of change of the surface area of a spherical balloon as it increases in volume is found using the derivative of the surface area formula with respect to time. However, without the balloon's radius value, it's impossible to give a numerical rate. Hence, we cannot select the correct option among the given choices without further information.
Step-by-step explanation:
The question involves finding the rate of change of the surface area of a sphere when we know the rate at which the volume is increasing. To find this, we need to use the formula for the volume of a sphere (V = 4/3πr^3) and the formula for the surface area of a sphere (SA = 4πr^2), where r is the radius of the sphere.
First, we find the rate of change of the radius with regard to time using the volume formula. Given that dV/dt = 25 cm³/sec, we differentiate both sides of the volume formula with respect to time (t) to get:
dV/dt = 4πr^2 * dr/dt
Therefore, we have:
25 = 4πr^2 * (dr/dt)
To find dr/dt, we would need the radius at the specific instant, but since the question doesn't provide that, we'll leave dr/dt as a function of r.
Next, we find the rate of change of the surface area (dSA/dt) using the formula for the surface area:
dSA/dt = 8πr * dr/dt
Substitute dr/dt from the previous step into this formula:
dSA/dt = 8πr * (25/(4πr^2))
Simplify to:
dSA/dt = (200/4r) cm²/sec = 50/r cm²/sec
The rate of change of surface area depends on r, so without the radius value, we can't give a numerical rate. Thus, we can't choose from options A, B, C, or D since we're missing essential information.