# Help with a volume question calculus style

1. Jun 4, 2004

### maccaman

Help with a volume question....calculus style

This is a problem solving question, so it is worded. I have cut off the first part which just explains the situation

We have a dam that opens to let water out to flood the plains. When a volume of 300 megalitres has passed through the gates, they automatically close. Within this volume limit, the rate of flow of water (R)
(in megalitres/minute) can be modelled by the following equation:

R = cos 0.001t, where t is the time in minutes.

If the gates open at midnight, at what time (to the nearest minute) will they automatically close?

This is causing me some considerable trouble, as i am only in grade 12. Any help with this would be greatly appreciated, thankyou

2. Jun 4, 2004

### AKG

Seeing as how you're in grade 12, this should be a cinch ;). It's a pretty simple problem, so I think most hints would give it away, but here it goes (let t=0 at midnight):

$$R = \cos (0.001t)$$

$$\frac{dV}{dt} = \cos (0.001t)$$

$$dV = \cos(0.001t) dt$$

$$\int_{V_0} ^{V_{f}} dV = \int_{0} ^{t_{f}} \cos(0.001t) dt$$

$$300 = 1000\sin (0.001t) |_{0} ^{t_{f}}$$

$$0.3 = \sin (0.001t_{f})$$

$$t_f = 1000 \arcsin (0.3)$$

Based on this you can figure out when it closes.

Last edited: Jun 4, 2004