Interpreting the derivative Find the derivati...

Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

Accepted Answer

Welcome back, everyone. A balloon is inflated and its volume in cubic meters after T seconds is given by the function V of T equals 4 T. Calculate V 10 and interpret the meaning of this value. Include units in your answer. So we're given our function the V of T, which is equal to 4 T, and the units would be cubic meters. So meters cubed, right? Essentially this gives us the volume of that balloon at any time T and time here is measured in seconds. What we want to do is evaluate V of T to begin with. So we want to differentiate. 4 T. and we want to factor out the constant, which is 4, so we're simply evaluating the derivative of T, which is DT divided by the T, and that's 1, so we get 4 for V of T. Now let's recall that whenever we differentiate the function. The derivative represents the rate of change. So we have got the rate of change function, and we have to understand that our rate of change is the rate of change of volume because we have differentiated the volume function. What we want to do for this problem is evaluate V at 10. But if we look at the expression of V of T, well, it doesn't have any t left, so there's a constant function, right? And we can say that whether we take 10 or any other t value in the given domain, that will always be 4, since the derivative is a constant function which is independent of time. Now what about the units? Well, rate of change is represented by the original units in the in this case meters cubed per time unit, and our time is measured in seconds, so that be meters cubed per second. And now knowing that V of T is the rate of change, we can write the final interpretation. So now we can say that the rate of change. The rate of change of. The volume Which is our original function. Off the balloon. We are including more specifics. Now, what is it? Well, it is for. Meters cupped per second and now when well at 10 seconds. So we add t equals 10. Seconds, because that's the point of interest, and we can label our final answer. The rate of change of the volume of the balloon is 4 m cubed per second at 10 seconds. Thank you for watching.

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.
When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

Key Concepts

Derivative

Physical Interpretation of Derivatives

Units of Measurement

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