Question #e6b1e

Dec 14, 2017

See below.

Explanation:

In order to solve this, we must find a function for the volume.

You are given:

$\frac{\mathrm{dV}}{\mathrm{dt}} = - 8 \left(25 - t\right)$

This is the derivative of some volume function, it is the function itself we require, so we must integrate this to get back the original function.

$- 8 \left(25 - t\right) = - 200 + 8 t$

$\int \left(- 200 + 8 t\right) \mathrm{dt} = - 200 t + 4 {t}^{2} + c$

We now need to find the value of $c$.

We are told that at the start the bucket contained 2.5 litres, this is equivalent to 2500 $c {m}^{3}$. This is at time $t = 0$

$\therefore$

$2500 = - 200 \left(0\right) + {\left(0\right)}^{2} + c$

So $c = 2500$

Our equation is now:

$V = - 200 t + 8 {t}^{2} + 2500$

We need to find the value of $t$ when the bucket is empty so:

$- 200 t + 8 {t}^{2} + 2500 = 0$

$t = 25$ and $t = 25$

I won't include the steps to solve this, as I'm sure you already know them.

So the time for the bucket to empty is:

$25 \textcolor{w h i t e}{}$ seconds.

Dec 14, 2017

$v = 4 {t}^{2} - 200 t + 2500$
$t = 25 \text{ s}$

Explanation:

It is given that

$\frac{\mathrm{dv}}{\mathrm{dt}} = - 8 \left(25 - t\right)$

Where $v$ is volume function.
Integrating both sides with respect to time $t$

$\int \frac{\mathrm{dv}}{\mathrm{dt}} \mathrm{dt} = \int \left(- 8 \left(25 - t\right)\right) \mathrm{dt}$
$\implies v = \int \left(- 200 + 8 t\right) \mathrm{dt}$
$\implies v = - 200 t + 4 {t}^{2} + C$ ......(1)
where $C$ is constant of integration.

To find the value of $C$ we make use initial condition. Given that at in the beginning bucket contained 2.5 litres of water. At $t = 0$
$v = 2.5 l = 2500 c {m}^{3}$.
Inserting these values in (1) we get

$2500 = - 200 \times 0 + {\left(0\right)}^{2} + C$
$\implies C = 2500$

Now (1) becomes

$v = - 200 t + 4 {t}^{2} + 2500$

rewriting it as

$v = 4 {t}^{2} - 200 t + 2500$ .....(2)

When the bucket is empty $v = 0$. Imposing the condition

$0 = 4 {t}^{2} - 200 t + 2500$
$\implies {t}^{2} - 50 t + 625 = 0$

This quadratic is a perfect square.

$\implies {\left(t - 25\right)}^{2} = 0$

Both roots are equal. Therefore, we have

$t = 25 \text{ s}$