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Classical accounts tell us that ...

Question

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Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour).

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the validity of the given statement. A car travels 480 miles in 8 hours. At some point, the car's speed exceeded 59 MPH. Awesome. So it appears for this particular problem, we're trying to figure out whether or not this provided statement is true for A or B false. So is it either a true or B false? So we're trying to ultimately determine whether or not a car travels 480 miles in 8 hours. At some point, the car's speed exceeded 59 MPH. So we're trying to determine if this statement is either true or false. So now that we know what we're ultimately trying to solve for, First off, let us note that we can check the validity of the statement by recalling and using the mean value theorem MVT for short. So we're trying to use, let's note the mean value theorem. So mean value theorem MVT for short. And we can, once again, we're trying to check the validity of this particular statement using our mean value theorem, and we need to note that since the falling conditions are met, we have two different conditions. One, the function is representing the car's distance traveled over time will be continuous over the closed interval of 0.8. That's what the square brackets mean. It's at a closed interval. So once again, the function is representing the car's distance traveled over time will be continuous over the closed interval of 0.8 in square brackets, and our second condition is the function is differentiable over the open interval 0.8. Awesome. So as we should be able to recall, the mean value theorem states that there is at least 1. C in the open interval A B in parentheses, such that F of C is equal to F of B minus F of A, all divided by B minus A. So in this particular context, FFT is the distance traveled at some time T, and F of T is the speed of the car at time T, and we're told that the total distance traveled is equal to 480 miles, and we're also told that the total time is equal to 8 hours. So, by the mean value theorem, there exists some C in the intervals 0.8, such that F C is equal to F 8 minus F of 0 divided by 8 minus 0. Which will be equal to 480 minus 0 divided by 8, which will be equal to 60 MPH. Fantastic. And since the average speed over the entire trip is 60 MPH, there must be at least one point during the trip where the car's speed was exactly 60 MPH. So therefore, without a doubt, the given, so given that the average speed exceeds 59 MPH, it implies at some point the car's instantaneous speed must have exceeded 59 MPH, and due to that logic, that means therefore, without a doubt, the final answer has to be multiple choice answer letter A, true. So once again, because the average speed exceeds 59 MPH as told to us by the problem itself, it implies that at some point the car's instantaneous speed must have exceeded 59 MPH, which means that our answer that we got for 60 MPH is good because that means we know that it can exceed 59 MPH, which ultimately makes our final answer A true. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Applications

Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour).

Key Concepts

Mean Value Theorem

Average Speed

Instantaneous Speed

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