When estimating distances from a table of velocity data, it is not necessary that the time intervals are equally spaced.

Question

When estimating distances from a table of velocity data, it is not necessary that the time intervals are equally spaced. After a space ship is launched, the following velocity data is obtained. Use these data to estimate the height above the Earth's surface at 122 seconds. Report answers accurate to 1 places. This is not meant to be a trick question...be careful of the UNITS!

t (sec)v (ft/s)
0	0
7	154
17	374
19	418
54	1188
60	1320
71	1837
122	4234

lower estimate of distance traveled: I got 37037 which is wrong. I multiplied the #s in the second column times 7 except for 4234.

upper estimate of distance traveled = 66675 which is also wrong. I multiplied all the #s in the 2nd column by 7 except 0.

DId I do my math wrong? what did I mess up on? My asnwer is suppose to be in miles so idk if I need to convert feet to miles? it also says to report my asnwer to 1 places...

Answer 1

Since the function is increasing over the interval, the left sum will be the lower estimate.

The left sum estimate would be 0(7-0) + 154(17 - 7) + 374(19 - 17) + 418(54 - 19) + 1188(60 - 54) + 1320(71 - 60) + 1837(122 - 71) which equals 132,253 ft

The right sum would be 154(7 - 0) + 374(17 - 7) + 418((19 - 17) + 1188(54 - 19) + 1320(60 - 54) + 1837(71 - 60) + 4234(122 - 71) which equals 291,295 ft

Neither of these is a very good estimate but the midpoint is not bad. That would be the same as averaging the left and right sums which equals 211,774 ft

To convert to miles, divide ft by 5,280 (since there are 5,280 ft per mile).

Answer 2

Hi Bri,

I hope this video answers your questions and helps you to visualize what's going on! Good luck!

Ross

Answer 3

The problem with multiplying each value by 7 is that the width of each of your rectangles is changing throughout the problem.

For instance, starting on the left side, from t: 0 to 7 you have a width of 7 and a left hand height of 0, so your first rectangle has an area of 0*7 = 0.

Continuing in this manner, your second rectangle goes from t = 7 to t = 17, so it has a width of 10 and a left hand height of 154, giving it an area of 10*154 = 1540.

If you go through the chart, each of your left hand rectangles should pan out this way with width first and height second:

7*0 + 10*154 + 2*374 + 35*418 + 6*1188 + 11*1320 + 51*1837 = 132,253 ft

(If you need to report in miles, then you should divide this by 5280)

For the right side, you'll do the same starting on the right: your first rectangle will have a width of 51 and a right hand height of 4234, so all together you'll have:

51*4234 + 11*1837 + 6*1320 + 35*1188 + 2*418 + 10*374 + 7*154 = 291,295 ft

Thanks!