2. Th e velocity of a car undergoing braking over the time interval [0, 6] is sh

Question

2. Th e velocity of a car undergoing braking over the time interval [0, 6] is shown below. v (ft/s) 75 50 25 t (seconds) 2 4 (a) Explain why the area function for the velocity function v(t) graphed here repre- sents the distance traveled by the car under braking. (Hint: why does the area of one Riemann rectangle re (b) Use a Riemann sum with six rectangles and right-side ordinates to estimate the total distance traveled by the car while braking o Eplain wy the anti derivai srced boe is no od here

Explanation / Answer

(a) In the Riemann rectangle method, we find the area under the curve by dividing the area into uniform figures rectangles. In the figure, we can divide the area under the curve into 6 Riemann rectangles.

Area of Riemann rectangle= time x velocity. But we know that velocity multiplied by time gives us the distance travelled. Therefore the area of one Riemann rectangle represents the distance travelled in that particular time frame.

The six Riemann rectangles are :  

Area of rectangle 1 > Area of rectangle 2 > Area of rectangle 3 >Area of rectangle 4>Area of rectangle 5>Area of rectangle 6

We observe that the area decreases with increase in time, which tells us that the distance decreases with increase in time. Since the distance travelled decreases with time and the velocity also decreases with time( ref the curve) the graprh represents the distance travelled by the car under braking.

b) Using Riemann sum with right ordinates the total distance travelled by the car while braking is given by total area or sum of areas of all the rectangles.
Total distance= sum of area of all rectangles= 75+50+50+37.5+25+12.5= 250 ft.

c) In the anti-derivative approach we take the integral of the function v(t) with respect to time over the limits time t=0 to t=6s. But this approachis no good here as we do not know the form of the function v(t) in terms of t.

Reimann Rectangle Breadth of recatangle Length of rectangle Area= l x b RECATANGLE 1 Time: 0 to 1 s vel: 75 ft/s Area: 75*1= 75ft RECATANGLE 2 Time: 1 to 2 s vel: 50 ft/s Area: 50*(2-1)=50ft RECATANGLE 3 Time: 2 to 3s vel: 50 ft/s Area: 50* (3-2)= 50ft RECATANGLE 4 Time: 3 to 4 s vel: 37.5 ft/s Area: 37.5ft RECATANGLE 5 Time: 4 to 5 s vel: 25 ft/s Area: 25 ft RECATANGLE 6 Time: 5 to 6 s vel: 12.5 ft/s Area: 12.5 ft

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