Question

Find the distance traveled in 14 seconds by an object that is moving with a velocity of v(t) = 11 + 6cos t feet per second.

A. 154.8204
B. 156.1704
C. 159.9436

Please show how this problem can be solved without derivatives and integrals, which we have not learned about yet.

Answer 1

Are you supposed to approximate it? although those answers look pretty close and accurate

Answer 2

Yes, it is supposed to be an approximation

Answer 3

using rectangles?

Answer 4

probably, although I'm not sure if we're supposed to use some sort of secant line

Answer 5

the cosine function oscillates between +1 and -1 , so if you think the velocity will be increasing and decreasing from 11+6 to 11- 6

Answer 6

The graph of cos(t) will have zero total area for each succession of pi radians... you see how cosine goes above and below the x axis, so each pi, the total area is zero ?

Answer 7

yes

Answer 8

So on the interval to t=14, it will complete 14/pi = 4.456 times, so up until 4pi maybe consider the velocity just a constant 11

Answer 9

So then estimate the area under cos(t) only for the last little bit, t=4pi = 12.566 to t=14

Answer 10

But then how do I get close to the numbers in the choices?

Answer 11

You can draw rectangles under the cos(t) function , the more the better, and the area of each of those is the distance traveled in that rectangle of time..

Answer 12

Okay, I will try that later. Thank you for the help!

Answer 13

v(t) = 11 + 6cos(t)
so the total distance can be aprox, the constant 11 for 14 sec
plus, the total approximation using rectangles of that last little bit

Answer 14

t=12.566 , 6*cos(t) = 6
t=14 , 6*cos(t) = 0.8204

Over that little time, the velocity goes from 6 to 0.8204,

Using just one rectangle half that height difference - 2.5897 and width t=(14 - 12.566) =1.43

The area here is about 1.43 * 2.5897 = 0.8182 distance

adding that to the constant 11 speed for 14 seconds

distance = 11 * 14 + 0.8182 = 154.81 total

Answer 15

THe value will get better if you divide the small chunk into more rectangles, and add their areas together, instead of using just 1 rectangle like above

Answer 16

Yes, it is more accurate once it's divided into several rectangles