The velocity ( mi \operatorname{mi} mi / h r hr h r ) of a drone flying in the air is given by v ( t ) = 12 + 4 t 2 v\left(t\right)=12+4t^2 v ( t ) = 12 + 4 t 2 for 0 ≤ t ≤ 4 0\le t\le4 0 ≤ t ≤ 4 hours. Let s ( 0 ) = 0 s\left(0\right)=0 s ( 0 ) = 0 . How far does the drone travel during the first hour?

been working on this problem where we have the velocity in miles per hour of a drone flying in the air given by this velocity function here. We're given this interval of time, and we're asked to let s of zero be equal to zero. And we wanna know now how far the drone is going to travel after the first hour. Now this was a function that we figured out in the previous video where we found the position as a function of time. Now since we're trying to figure out in this part how far the drone travels in the first hour, well, what we can simply do is recognize that we just need to plug in one hour for our time. So what that means is we're looking for s of one, which is going to be four times one cubed over three plus 12 times one. Now that's going to be the same thing as four over three plus 12. And to write this as a decimal, you should get an approximate value of 13.3 meters. Now on a calculator, this would come out to 13.333, and this would just keep on going. Let's go ahead and round this off. I'm just gonna say about 13.3 meters, and that is how far our drone has traveled after the first hour. So that's the solution to part b. Let's move on to the next video.

10. Physics Applications of Integrals

Kinematics

Calculus

10. Physics Applications of Integrals

Kinematics

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Multiple Choice

The velocity ( $\operatorname{mi}$ / $hr$ ) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le4$ hours. Let $s\left(0\right)=0$ .
How far does the drone travel during the first hour?

13.3 mi

133.3 mi

8.00 mi

6.67 mi

Verified step by step guidance

Step 1: Recognize that the problem involves finding the distance traveled by the drone during the first hour. The distance traveled is the integral of the velocity function over the given time interval.

Step 2: Write the velocity function as v(t) = 12 + 4t^2. The distance traveled, s(t), can be found by integrating v(t) with respect to t over the interval [0, 1].

Step 3: Set up the definite integral for the distance: s(1) = ∫[0 to 1] (12 + 4t^2) dt.

Step 4: Break the integral into two parts for easier computation: ∫[0 to 1] 12 dt + ∫[0 to 1] 4t^2 dt. Compute each term separately. For the first term, integrate 12 with respect to t. For the second term, integrate 4t^2 with respect to t.

Step 5: Evaluate the definite integrals by substituting the limits of integration (0 and 1) into the antiderivatives. Add the results of the two integrals to find the total distance traveled during the first hour.