Question

a faucet is turned on at 9:00am and water starts to flow into a tank at the rate of r(t)=7t√, where t is time in hours after 9:00am and the rate r(t) is in cubic feet per hour.

Answer 1

The water volume,
`\( V(t) \)`, that flows into the tank from 9:00am onwards is given by
`\( V(t) = (7)/(2)√(t^3) \)` cubic feet.

Step-by-step explanation:

The rate of water flow,
`\( r(t) = 7t√(t) \)`, represents the derivative of the volume function, V(t). To find V(t) , we integrate r(t) with respect to t. The antiderivative of
`\( r(t) \) is \( (7)/(2)√(t^3) \)`. Therefore, the volume function is
`\( V(t) = (7)/(2)√(t^3) + C \)`, where C is the constant of integration.

Since the water starts flowing at 9:00am, t = 0 at that moment. Substituting this into the volume function, we can determine the value of C to be zero. Thus, the final volume function is
`\( V(t) = (7)/(2)√(t^3) \)` cubic feet.

This volume function gives the total amount of water that has flowed into the tank up to time t after 9:00am. It is crucial to note that the constant of integration is zero, as the volume is zero at t = 0 when the water flow begins.

Answer 2

To calculate the flow rate of water from a faucet, use the formula Q = A⋅v. The diameter of the stream below the faucet remains the same if the flow velocity is unchanged and effects such as surface tension are neglected.

Step-by-step explanation:

The question involves calculating the flow rate of water from a faucet and understanding how the characteristics of the flow change due to gravity. To find the flow rate in cubic centimeters per second (cm³/s), you'd use the formula Q = A⋅v, where A is the cross-sectional area of the faucet and v is the velocity of the flow. For a faucet with a diameter of 1.80 cm and water emerging at a speed of 0.500 m/s, one must convert the speed to cm/s and apply the formula to find Q. Regarding part (b) of the question, due to the principle of conservation of mass and assuming that the speed does not change with distance from the faucet, the diameter of the stream would remain the same 0.200 m below the faucet. In both problems, we neglect any effects due to surface tension or viscosity which could alter the flow characteristics in real-life scenarios.