A cylindrical tank with radius 7 m is being filled with water at a rate of 2 m3/min. How fast is the height of the water increasing (in m/min)?
Question
A cylindrical tank with radius 7 m is being filled with water at a rate of 2 m3/min. How fast is the height of the water increasing (in m/min)?
Solution
The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
Given that the radius r is 7 m and the volume V is increasing at a rate of 2 m³/min, we want to find dh/dt, the rate at which the height h is increasing.
We can start by differentiating both sides of the volume formula with respect to time t to get dV/dt = πr²dh/dt.
We can then solve for dh/dt to get dh/dt = dV/dt / (πr²).
Substituting the given values into this equation gives dh/dt = 2 m³/min / (π*(7 m)²) = 2 m³/min / (153.9380400259 m²) = 0.013 m/min.
So, the height of the water is increasing at a rate of 0.013 m/min.
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