Answer:
[tex]\displaystyle \frac{dh}{dt}=-\frac{4}{\pi}\approx-1.2732\text{ centimeters per minute}[/tex]
The water level is dropping by approximately 1.27 centimeters per minute.
Step-by-step explanation:
Please refer to the attached diagram.
The height of the conical container is 6 cm, and its radius is 1 cm.
The container is leaking water at a rate of 1 cubic centimeter per minute.
And we want to find the rate at which the water level h is dropping when the water height is 3 cm.
Since we are relating the water leaked to the height of the water level, we will consider the volume formula for a cone, given by:
[tex]\displaystyle V=\frac{1}{3}\pi r^2h[/tex]
Now, we can establish the relationship between the radius r and the height h. At any given point, we will have two similar triangles as shown below. Therefore, we can write:
[tex]\displaystyle \frac{1}{6}=\frac{r}{h}[/tex]
Solving for r yields:
[tex]\displaystyle r=\frac{1}{6}h[/tex]
So, we will substitute this into our volume formula. This yields:
[tex]\displaystyle \begin{aligned} V&=\frac{1}{3}\pi \Big(\frac{1}{6}h\Big)^2h\\ &=\frac{1}{108}\pi h^3\end{aligned}[/tex]
Now, we will differentiate both sides with respect to time t. Hence:
[tex]\displaystyle \frac{d}{dt}[V]=\frac{d}{dt}\Big[\frac{1}{108}\pi h^3\Big][/tex]
The left is simply dV/dt. We can move the coefficient from the right:
[tex]\displaystyle \frac{dV}{dt}=\frac{1}{108}\pi\frac{d}{dt}\big[h^3\big][/tex]
Implicitly differentiate:
[tex]\displaystyle\begin{aligned} \frac{dV}{dt}&=\frac{1}{108}\pi(3h^2\frac{dh}{dt})\\ &=\frac{1}{36}\pi h^2\frac{dh}{dt}\end{aligned}[/tex]
Since the water is leaking at a rate of 1 cubic centimeter per minute, dV/dt=-1.
We want to find the rate at which the water level h is dropping when the height of the water is 3 cm.. So, we want to find dh/dt when h=3.
So, by substitution, we acquire:
[tex]\displaystyle -1=\frac{1}{36}\pi(3)^2\frac{dh}{dt}[/tex]
Therefore:
[tex]\displaystyle -1=\frac{1}{4}\pi\frac{dh}{dt}[/tex]
Hence:
[tex]\displaystyle \frac{dh}{dt}=-\frac{4}{\pi}\approx-1.2732\text{ centimeters per minute}[/tex]
The water level is dropping at a rate of approximately 1.27 centimeters per minute.
