Answer:
The water level is rising at a rate of approximately 0.1415 meters per minute.
Step-by-step explanation:
Water is flowing into a right cylindrical-shaped swimming pool at a rate of 4 cubic meters per minute. The radius of the base is 3 meters.
And we want to determine the rate at which the water level of the pool is rising.
Recall that the volume of a cylinder is given by:
[tex]\displaystyle V = \pi r^ 2h[/tex]
Since the radius is a constant 3 meters:
[tex]\displaystyle V = 9\pi h[/tex]
Water is flowing at a rate of 4 cubic meters per minute. In other words, dV/dt = 4 m³ / min.
Take the derivative of both sides with respect to t:
[tex]\displaystyle \frac{d}{dt}\left[ V\right] = \frac{d}{dt}\left[ 9\pi h\right][/tex]
Implicitly differentiate:
[tex]\displaystyle \frac{dV}{dt} = 9\pi \frac{dh}{dt}[/tex]
The rate at which the water level is rising is represented by dh/dt. Substitute and solve:
[tex]\displaystyle \left(4 \right) = 9\pi \frac{dh}{dt}[/tex]
Therefore:
[tex]\displaystyle \frac{dh}{dt} = \frac{4}{9\pi} \approx 0.1415\text{ m/min}[/tex]
In conclusion, the water level is rising at a rate of approximately 0.1415 meters per minute.
