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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.

Sagot :

MrRoyalidn

Answer:

(a)

i) [tex]V=12.6m/s[/tex]

ii) [tex]V=12.06m/s[/tex]

iii) [tex]V=12.006m/s[/tex]

(b)

[tex]V = 10m/s[/tex]

Step-by-step explanation:

Given

[tex]s = 6t^2 + 4[/tex]

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

When t = 1

[tex]t_1 = 1[/tex]

[tex]s_1 = 6t^2 + 4 = 6 * 1^2 + 4 = 6 + 4 =10[/tex]

i) h = 0.1

When t = 1 + h

[tex]t_2 = 1 + 0.1 = 1.1[/tex]

[tex]s_2= 6t^2 + 4 = 6 * (1.1)^2 + 4 = 11.26[/tex]

Average velocity is then calculated as:

[tex]V = \frac{s_2 - s_1}{t_2 - t_1}[/tex]

[tex]V = \frac{11.26 - 10}{1.1- 1} = \frac{1.26}{0.1} = 12.6[/tex]

[tex]V=12.6m/s[/tex]

ii) h = 0.01

When t = 1 + h

[tex]t_2 = 1 + 0.01 = 1.01[/tex]

[tex]s_2= 6t^2 + 4 = 6 * (1.01)^2 + 4 = 10.1206[/tex]

Average velocity is then calculated as:

[tex]V = \frac{s_2 - s_1}{t_2 - t_1}[/tex]

[tex]V = \frac{10.1206 - 10}{1.01- 1} = \frac{0.1206}{0.01} = 12.06[/tex]

[tex]V=12.06m/s[/tex]

ii) h = 0.001

When t = 1 + h

[tex]t_2 = 1 + 0.001 = 1.001[/tex]

[tex]s_2= 6t^2 + 4 = 6 * (1.001)^2 + 4 = 10.012006[/tex]

Average velocity is then calculated as:

[tex]V = \frac{s_2 - s_1}{t_2 - t_1}[/tex]

[tex]V = \frac{10.012006 - 10}{1.001- 1} = \frac{0.012006 }{0.001} = 12.006[/tex]

[tex]V=12.006m/s[/tex]

Solving (b): Instantaneous velocity at t = 1

When t = 1

[tex]t = 1[/tex]

[tex]s = 6t^2 + 4 = 6 * 1^2 + 4 = 6 + 4 =10[/tex]

Velocity is:

[tex]V = \frac{s}{t}[/tex]

[tex]V = \frac{10}{1}[/tex]

[tex]V = 10m/s[/tex]

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