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Calculus

# Related Rates - 3D Geometry

The radius of a sphere increases by $$1$$ cm per second. If $$V$$ is the volume of the sphere and $$t$$ is time measured in seconds, what is $$\frac{dV}{dt}$$ (in cm/s) for $$r= 13$$ ?

The side length $$x$$ of a cube increases by $$2$$ cm per minute. If $$V$$ denotes the volume of the cube in $$\text{cm}^3$$, and $$t$$ denotes time measured in minutes, what is the change in volume over time $$\frac{dV}{dt}$$ (in $$\text{ cm}^3/\text{min}$$) for $$x=6$$ cm?

The radius $$r$$ of a sphere increases by $$1$$ cm per second. If $$S$$ denotes the surface area of the sphere, and $$t$$ denotes time measured in seconds, what is $$\frac{dS}{dt}$$ (in $$\text{ cm}^2/\text{s}$$) for $$r= 9 \text{ cm}$$?

A spherical fish tank of radius 10 meters is being filled with at a constant rate of $$R\pi$$ cubic meters of water per second. At height 6 meters, the rate of change of the height is 1 meter per second. What is the value of $$R$$?

Details and assumptions

The height is measured from the bottom of the sphere (i.e. in contact with the ground).

A circular conical reservoir, vertex down, has depth $$14\text{ ft}$$ and top radius $$7\text{ ft}.$$ Water is leaking out at the rate of $$\frac{1}{3}$$ cubic feet per second. Find the rate at which the water level is dropping when the diameter of the surface is $$7\text{ ft}.$$

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