Question

Water is flowing into a large spherical tank at a constant rate. Let V (t) be the volume of water in the tank at time t, and h(t) be the height of the water level at time t.

a) Give a physical interpretation of dV/dt and dh/dt.
b) Which of dV dt and dh dt is constant? Explain your answer.
c) Is dV dt positive, negative, or zero when the tank is one quarter full?
d) Is dh dt positive, negative, or zero when the tank is one quarter full?

Answer 1

See solutions for detail.

Explanation:

a.
`(dV)/(dt)` is the instantaneous rate of change of volume given with respect to time, t.

The volume's rate of change is written as a function of time.

-
`(dh)/(dt)` is the rate of change in the height of water in the tank with respect to time, t.

b.
`(dV)/(dt)-` is the only constant. Water flows into the constant at a constant rate, say
`6cm^3` per minute.

c.
`(dV)/(dt)` is positive. Volume water in the take is increasing from time to time.

-The volume at time t=1 is greater than the volume at t=0, hence, it's a positive rate of change.

d.
`(dh)/(dt)` is a positive rate. The initial height of water in the tank is zero.

-The final height at time t is 0.25h. The height is increasing with time.

Hence, it is positive.