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In a time of t seconds, a particle moves a distance of s meters from its starting point, where s = 6t^2 + 4.

(a) Find the average velocity between t = 1 and t = 1 + k if (i) h = 0.1 (ii) h = 0.01 (iii) h = 0.001 Enter the exact answers.
(i) When h = 0.1, the average velocity between t = 1 and t = 1 + h is m/sec.
(ii) When h = 0.01, the average velocity between t = 1 and t = 1 + h is m/sec.
(iii) When h = 0.001, the average velocity between t = 1 and t = 1 + h is m/sec.
(b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time t = 1. Round your estimate to the nearest integer. The instantaneous velocity appears to be m/sec.

User Gbarry
by
4.8k points

1 Answer

12 votes

Answer:

(a)

i)
V=12.6m/s

ii)
V=12.06m/s

iii)
V=12.006m/s

(b)


V = 10m/s

Explanation:

Given


s = 6t^2 + 4

Solving (a): Average velocity between t = 1 and t = 1 + h

When t = 1


t_1 = 1


s_1 = 6t^2 + 4 = 6 * 1^2 + 4 = 6 + 4 =10

i) h = 0.1

When t = 1 + h


t_2 = 1 + 0.1 = 1.1


s_2= 6t^2 + 4 = 6 * (1.1)^2 + 4 = 11.26

Average velocity is then calculated as:


V = (s_2 - s_1)/(t_2 - t_1)


V = (11.26 - 10)/(1.1- 1) = (1.26)/(0.1) = 12.6


V=12.6m/s

ii) h = 0.01

When t = 1 + h


t_2 = 1 + 0.01 = 1.01


s_2= 6t^2 + 4 = 6 * (1.01)^2 + 4 = 10.1206

Average velocity is then calculated as:


V = (s_2 - s_1)/(t_2 - t_1)


V = (10.1206 - 10)/(1.01- 1) = (0.1206)/(0.01) = 12.06


V=12.06m/s

ii) h = 0.001

When t = 1 + h


t_2 = 1 + 0.001 = 1.001


s_2= 6t^2 + 4 = 6 * (1.001)^2 + 4 = 10.012006

Average velocity is then calculated as:


V = (s_2 - s_1)/(t_2 - t_1)


V = (10.012006 - 10)/(1.001- 1) = (0.012006 )/(0.001) = 12.006


V=12.006m/s

Solving (b): Instantaneous velocity at t = 1

When t = 1


t = 1


s = 6t^2 + 4 = 6 * 1^2 + 4 = 6 + 4 =10

Velocity is:


V = (s)/(t)


V = (10)/(1)


V = 10m/s

User Apolonia
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4.9k points