Calculus

Related Rates

Related Rates - 3D Geometry

         

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

The side length xx of a cube increases by 22 cm per minute. If VV denotes the volume of the cube in cm3\text{cm}^3, and tt denotes time measured in minutes, what is the change in volume over time dVdt \frac{dV}{dt} (in  cm3/min\text{ cm}^3/\text{min} ) for x=6x=6 cm?

The radius rr of a sphere increases by 1 1 cm per second. If SS denotes the surface area of the sphere, and tt denotes time measured in seconds, what is dSdt\frac{dS}{dt} (in  cm2/s\text{ cm}^2/\text{s}) for r=9 cmr= 9 \text{ cm}?

A spherical fish tank of radius 10 meters is being filled with at a constant rate of RπR\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 RR?

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 ft14\text{ ft} and top radius 7 ft.7\text{ ft}. Water is leaking out at the rate of 13\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 ft.7\text{ ft}.

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