Question

A model rocket has upward velocity v(t) = 10t2 ft/s, t seconds after launch. Use the interval [0, 6] with n = 6 and equal subintervals to compute the following approximations of the distance the rocket traveled. (Round your answers to two decimal places.

(a) Left-hand sum = _____ ft
(b) Right-hand sum = _____ ft
(c) average of the two sums = ______ ft

Answer 1

a)550

b)910

c)730

Explanation:

The given model is

`v(t) = 10t^2 ft/s`

Use the interval [0,6], with n=6 rectangles

Then, the interval width is

`\Delta t = (b-a)/(n)`

`\Delta t = (6-0)/(6)`= 1

so, the sub intervals are

[0,1], [1,2], [2,3], [3,4],[4,5],[5,6]

Now evaluating the function values

`f(t_0)= f(0) = 0`

`f(t_1)= f(1) = 10`

`f(t_2)= f(2) = 40`

`f(t_3)= f(3) = 90`

`f(t_4)= f(4) = 160`

`f(t_5)= f(5) = 250`

`f(t_6)= f(6) = 360`

a) left hand sum is

L_6 =
`\Delta t [f(t_0)+ f(t_1)+f(t_2)+f(t_3)+f(t_4)+f(t_5)]`

=
`1 [0+ 10+40+90+160+250]`

= 550

b) right hand sum

R_6 =
`\Delta t [ f(t_1)+f(t_2)+f(t_3)+f(t_4)+f(t_5)+f(t_6)]`

=
`1 [10+40+90+160+250+360]`

= 910

c) average of two sums is

`(L_5+R_5)/(2)`

=
`(550+910)/(2)`

=730