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water flows into a cylindrical tank at the rate of 2 ft/min. if the radius of the top of the tank is 4ft and the height is 6 ftdetermine how quickly the water level is rising when the water is 2 feet deep in the tank.

Sagot :

0.477 ft/min, the water level is rising when the water is 2 feet deep in the tank.

What is Derivative?

The derivative of a function of a real variable in mathematics quantifies the sensitivity of the function's value (output value) to changes in its argument (input value).

Calculus's core tool is the derivative. The velocity of an item, for instance, is the derivative of its position with respect to time; it quantifies how quickly the object's position varies as time passes.

When it occurs, the slope of the tangent line to the function's graph at a given input value is the derivative of a function of a single variable.

The function closest to that input value is best approximated linearly by the tangent line. Because of this, the derivative is

According to our question-

The tank's water volume, V, is as follows:

V=πr2h

When the height, h, of the water rising from the tank's bottom increases with time, t, and the radius, r=4ft, is constant.

Using the volume's first derivative, we obtain:

dVdt=πr2dhdt

How to solve for the height change over time:

dhdt=1πr2dVdt

dhdt=1π(4ft)2(24ft3min)

dhdt≈0.477ftmin

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