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How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.

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).

ConicalReservoir

ConicalReservoir

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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