Help with a volume question calculus style

[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 modeled 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

 

[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.

 

[M98Ranger]

To solve this problem, we first need to understand that the volume of water that passes through the gates is equal to the area under the curve of the flow rate function. In other words, we need to find the time interval where the total area under the curve is equal to 300 megalitres.

To do this, we can use the fundamental theorem of calculus which states that the integral of a function over a certain interval is equal to the difference between the function evaluated at the upper and lower limits of the interval. In this case, we need to find the time interval where the integral of the flow rate function is equal to 300 megalitres.

So, let's set up our integral:

∫cos 0.001t dt = 300

To solve this, we can use a calculator or a computer to evaluate the integral. This gives us:

(1000/0.001)sin 0.001t = 300

sin 0.001t = 0.3

Now, we need to find the time (t) when the sine function equals 0.3. We can use a calculator to find the inverse sine of 0.3, which is approximately 17.46 degrees. To convert this to minutes, we need to multiply it by 60 since the time is measured in minutes, not degrees. So, the time interval where the gates will automatically close is approximately 17.46 * 60 = 1047.6 minutes.

Therefore, the gates will automatically close at around 1047.6 minutes after midnight, which is approximately 5:48 AM. Keep in mind that this is an approximation and may not be the exact time, but it should be very close.

I hope this helps you understand how to solve this problem using calculus. If you have any further questions, feel free to ask. Good luck!