Rates of Change

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Question

Rates of Change

Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says at 10 minutes after being inflated, the volume of a balloon is given by VFT, which is equal to 40 cubed centimeters cubed. What is the rate of change in the volume of the balloon 5 minutes after it starts inflating? So we're asked in this problem to find the rate of change in the volume of the balloon. And so we're looking for the change in volume divided by our change in time, which we'll label as Delta V divided by delta T. We're actually interested in the instantaneous rate of change. So, we call to find the instantaneous rate of change at a particular point, that is going to be the limit, as H approaches 0, of the quantity of V of T + H. Minus VFT. In quantity divided by each. So we're interested in the time, which is 5 minutes after the balloon starts inflating. So that means that T is going to be equal to 5. So, now, if we look at our expression, we'll have Delta V divided by delta T. It's equal to the limit, as H approaches 0 of the quantity, and here we'll substitute in 5 for T. So we'll have V of 5 plus H. Minus V 5 in quantity, divided by H. So we need to evaluate the expressions for V 5 plus H and VF 5. And we can do that using the formula that we were given in the problem for VFT. So we'll start off by looking at V 5 plus H. And this is going to be equal to 4, multiplied by the quantity of 5 plus H in quantity cut. So multiplying out the binomial, this is going to be equal to 4, multiplied by the quantity of 125 plus 75 H plus 15 squared plus H cued in quantity. And now distributing the 4, this is going to be equal to 500. Plus 300 each. Plus 60 8 squared. Plus 4 H cubed. So now we need to evaluate V of 5, and this is going to be equal to 4 multiplied by 5 cubed. Which is equal to 500. So, now we can substitute these expressions back into our rate of change formula. So that means that Delta V divided by delta T, is going to be equal to the limit, as H approaches 0, of the quantity of 500 plus 300 H. Plus 60 squared plus 4 cubed minus 500. In quantity and that is divided by h. So, here, if we look at the numerator, we have 500 minus 500, so when I combine those two terms, I get 0, and the remaining terms all have a factor of H in them. So I can divide out one factor of H out of each of those terms with the h in the denominator. So doing that, this is going to be equal to the limit, as H approaches 0 of the quantity of 300 plus 60 H + 4 H squad. So we can evaluate this limit by direct substitution. So setting H equal to 0, this is going to be equal to 300 plus 60 multiplied by 0 plus 4 multiplied by 0 squared. So evaluating this expression, this is going to be equal to 300. And if we look at units for our answer here, we see that volume was given in centimeters cubed and time was given in minutes. So that means the units for our answer here are going to be centimeters cubed per minute. So the final answer for this problem is 300 centimeters cubed per minute. So I hope that this explanation has been useful, and I'll see you in the next video.

Rates of Change

Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?

Key Concepts

Derivative

Function of Time

Application of the Chain Rule

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