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A particle moves along the
x-axis so that at time
t >= 0 its velocity is given by
v(t)=6t^(2)-48 t+90. Determine all intervals when the speed of the particle is decreasing.

A particle moves along the $x$ -axis so that at time $t \geq 0$ its velocity is given by $v(t)=6 t^{2}-48 t+90$ . Determine all intervals when the speed of the particle is decreasing.

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=6 t^{2}-48 t+90

. Determine all intervals when the speed of the particle is decreasing.

Find Acceleration Function: To find when the speed of the particle is decreasing, we need to find when the derivative of the velocity, which is the acceleration, is negative. This is because if the acceleration is negative, the velocity is decreasing, and hence the speed is decreasing as well. $\newline$ First, we need to find the acceleration by taking the derivative of the velocity function $v(t) = 6t^2 - 48t + 90$ . $\newline$ $a(t) = \frac{dv}{dt} = \frac{d}{dt} (6t^2 - 48t + 90)$ $\newline$ $a(t) = 12t - 48$
Determine Negative Acceleration: Next, we need to determine when the acceleration $a(t)$ is negative, as this will indicate when the speed is decreasing. $\newline$ We set the acceleration function less than zero and solve for $t$ : $\newline$ 12t - 48 < 0 $\newline$ 12t < 48 $\newline$ t < 4
Check Endpoints for Speed: The inequality t < 4 means that the acceleration is negative and hence the speed is decreasing for all times $t$ in the interval $[0, 4)$ . However, we must check the endpoints to ensure that the speed is indeed decreasing up to $t = 4$ and not just approaching it.
Verify Speed Decrease: We check the velocity at $t = 0$ and $t = 4$ to see if there is a change in sign which would indicate a maximum or minimum, and hence a change in the direction of the speed. $v(0) = 6(0)^2 - 48(0) + 90 = 90$ $v(4) = 6(4)^2 - 48(4) + 90 = 6(16) - 192 + 90 = 96 + 90 - 192 = -6$ The velocity decreases from $90$ at $t = 0$ to $-6$ at $t = 4$ , confirming that the speed is decreasing in the interval $[0, 4)$ .

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