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Answer is Right!
Answer is Wrong!
The correct answer is $\boxed{\text{C}}$.
The acceleration of a particle is given by $a = \frac{d^2x}{dt^2}$. In this case, we have $x = t^3 – 3t^2 + 5$, so $a = 3t^2 – 6t$.
The acceleration after 5 seconds is $a(5) = 3(5)^2 – 6(5) = 15$.
The acceleration after 3 seconds is $a(3) = 3(3)^2 – 6(3) = 9$.
The ratio of the accelerations is $\frac{a(5)}{a(3)} = \frac{15}{9} = \boxed{5}$.
Here is a brief explanation of each option:
- Option A: $2$. This is the ratio of the velocities after 5 seconds and 3 seconds. The velocity of a particle is given by $v = \frac{dx}{dt}$. In this case, we have $x = t^3 – 3t^2 + 5$, so $v = 3t^2 – 6t$. The velocity after 5 seconds is $v(5) = 3(5)^2 – 6(5) = 25$. The velocity after 3 seconds is $v(3) = 3(3)^2 – 6(3) = 9$. Therefore, the ratio of the velocities is $\frac{v(5)}{v(3)} = \frac{25}{9} \neq 2$.
- Option B: $3$. This is the ratio of the positions after 5 seconds and 3 seconds. The position of a particle is given by $x = t^3 – 3t^2 + 5$. In this case, we have $x = t^3 – 3t^2 + 5$, so $x(5) = 5^3 – 3(5)^2 + 5 = 75$. The position after 3 seconds is $x(3) = 3^3 – 3(3)^2 + 5 = 14$. Therefore, the ratio of the positions is $\frac{x(5)}{x(3)} = \frac{75}{14} \neq 3$.
- Option C: $4$. This is the ratio of the accelerations after 5 seconds and 3 seconds. As shown above, the acceleration after 5 seconds is $a(5) = 15$ and the acceleration after 3 seconds is $a(3) = 9$. Therefore, the ratio of the accelerations is $\frac{a(5)}{a(3)} = \frac{15}{9} = 5$.
- Option D: $5$. This is the ratio of the velocities after 5 seconds and 3 seconds. As shown above, the velocity after 5 seconds is $v(5) = 25$ and the velocity after 3 seconds is $v(3) = 9$. Therefore, the ratio of the velocities is $\frac{v(5)}{v(3)} = \frac{25}{9} \neq 5$.