The radius r of a sphere is increasing at a rate of 8 inches per minute. (a) Find the rate of change of the volume when r = 8 inches. in.3/min (b) Find the rate of change of the volume when r = 38 inches. in.3/min

a) 2421.743 in³/ min , b) 54437.52 in³/ min

Explanation: The formulae for the volume of a sphere is given below as

V = (4π/3) ×r³ where V =volume of sphere and r = radius of sphere.

By taking the time derivative of the formulae, we have the rate of change of volume with time (we do so by using implicit diffrenciation) , hence we have that

dV/dt = (4π/3) × 3r²×dr/dt

Where dV/dt = rate of change of volume and dr/dt= rate of change of radius.

A)

r = 8 inches, dr/dt = 3 in/min

By substituting this into the formulae, we have that

dV/dt = (4π/3) × 3r²×dr/dt

dV/dt = (4π/3) × 3(8)² × 3

dV/dt = (4π/3) × 576

dV/dt = 2421.743 in³/ min

B)

r = 38 inches, dr/dt = 3 in/min

By substituting this into the formulae, we have that

dV/dt = (4π/3) × 3r²×dr/dt

dV/dt = (4π/3) × 3(38)² × 3

dV/dt = (4π/3) × 12996

dV/dt = 54437.52 in³/ min

the minimum constant speed at which she can catch the train is 2.7 m/s

given:

acceleration = 0.6

time = 4.5 s

to find:

velocity = ?

formula used:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

solution:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

v = 0 + (0.6 × 4.5)

v = 2.7 m/s

the minimum constant speed at which she can catch the train is 2.7 m/s

i think the answer is a and b

i wish you luck