Please note that someone already showed me the upper, lower and middle bounds fo

Question

Please note that someone already showed me the upper, lower and middle bounds for this, but when I tried following their work, none of the answers were accepted

A student is speeding down Route 11 in his fancy red Porsche when his radar system warns him of an obstacle 400 feet ahead. He immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of him. The "black box" in the Porsche records the car's speed every two seconds, producing the following table. The speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a uniform rate.

Time since brakes applied (sec) 0 2 4 6 8 10
Speed (ft/sec) 105 85 50 30 15 0

A. What is your best estimate of the total distance the student's car traveled before coming to rest (note that the best estimate is probably not the over or under estimate that you can most easily find)?
distance = ________? (include units)

Explanation / Answer

The whole point of differential calculus is to find the quantity of something at an instant in time. This is how/why Newton invented differential calculus. Plot a graph of the speed versus time. If you have Excel use chart wizard to plot a scatter plot of speed on the y axis versus time on the x axis. Look at the slope of the lines between each point. That is an estimation of the acceleration (negative acceleration or "deceleration" in this problem) Use the formula: s = Vot + ½at² to find the distance between each segment of line and then add them all up. For the first one: Vo = 120ft/sec, t = 2 sec and a would be -25 ft/sec² a from the graph is the slope (first derivative) of the line connecting the first two points so... slope of line = m = Y2-Y1/X2-X1 = 70-120/2 = -50/2 = -25ft/sec² so a = -25 ft/sec² Now you work the remaining 4 intervals and remember, to finally add up all of these distances to get the total distance. In real life that black box would be taking several million readings a second and you'd have a smooth function that would be differentiable.

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