Mathematics

Problem

Oil drains from a storage tank according to

V(t) = 2000 - 50t^{2}

, where

V(t)

is the volume (in gallons) remaining

t

hours after a valve is opened. Determine when the oil is leaving the tank at a rate of

600\,\text{gal/h}

and describe how that rate is changing at that instant. a) At

t \approx 5.3\,\text{h}

, the oil is leaving at

600\,\text{gal/h}

and the drain rate is becoming faster because

V''(5.3) = -100\,\text{gal/h}^2

. b) At

t = 12\,\text{h}

, the oil is leaving at

600\,\text{gal/h}

and the drain rate is becoming faster because

V''(12) = -50\,\text{gal/h}^2

. c) At

t = 6\,\text{h}

, the oil is leaving at

600\,\text{gal/h}

and the drain rate is becoming faster because

V''(6) = -100\,\text{gal/h}^2

. d) At

t = 6\,\text{h}

, the oil is leaving at

600\,\text{gal/h}

and the drain rate is becoming slower because

V''(6) = +100\,\text{gal/h}^2