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A particle travels along the
x-axis such that its velocity is given by
v(t)=t^(0.3)-4cos(2t). What is the acceleration of the particle at time
t=5? You may use a calculator and round your answer to the nearest thousandth.
Answer:

A particle travels along the $x$ -axis such that its velocity is given by $v(t)=t^{0.3}-4 \cos (2 t)$ . What is the acceleration of the particle at time $t=5 ?$ You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{0.3}-4 \cos (2 t)

. What is the acceleration of the particle at time

t=5 ?

You may use a calculator and round your answer to the nearest thousandth.

\newline

Answer:

Identify Relationship: Identify the relationship between velocity and acceleration. Acceleration is the derivative of velocity with respect to time. To find the acceleration at a specific time, we need to differentiate the velocity function $v(t)$ with respect to $t$ .
Differentiate Velocity: Differentiate the velocity function $v(t) = t^{0.3} - 4\cos(2t)$ with respect to time $t$ to find the acceleration function $a(t)$ . Using the power rule, the derivative of $t^{0.3}$ is $0.3 \cdot t^{-0.7}$ . Using the chain rule, the derivative of $-4\cos(2t)$ is $8\sin(2t)$ . So, $a(t) = 0.3 \cdot t^{-0.7} + 8\sin(2t)$ .
Evaluate Acceleration: Evaluate the acceleration function $a(t)$ at $t=5$ . Substitute $t$ with $5$ in the acceleration function $a(t) = 0.3 \cdot t^{-0.7} + 8\sin(2t)$ . $a(5) = 0.3 \cdot 5^{-0.7} + 8\sin(2\cdot 5)$ .
Compute Values: Use a calculator to compute the values. $\newline$ $a(5) = 0.3 \times 5^{-0.7} + 8\sin(10)$ . $\newline$ Using a calculator, we find: $\newline$ $5^{-0.7} \approx 0.1749$ (rounded to four decimal places). $\newline$ $\sin(10) \approx -0.5440$ (rounded to four decimal places). $\newline$ Now, calculate the acceleration: $\newline$ $a(5) \approx 0.3 \times 0.1749 + 8 \times (-0.5440)$ . $\newline$ $a(5) \approx 0.05247 - 4.352$ . $\newline$ $a(5) \approx -4.29953$ .
Round Acceleration: Round the acceleration to the nearest thousandth. $\newline$ $a(5) \approx -4.300$ (rounded to three decimal places).

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