Question

the position of an object is given as a function of time by x = 3t² + 5t³ - 2t. what is the velocity of the object at time t = 2 s? 72 m/s 66 m/s 64 m/s 70 m/s

Explanation:

Step1: Recall the formula for velocity

Velocity \(v=\frac{dx}{dt}\). Given \(x = 3t^{2}+5t^{3}-2t\), use the power rule \(\frac{d}{dt}(at^{n})=n\times at^{n - 1}\).

Step2: Differentiate \(x\) with respect to \(t\)

\(\frac{dx}{dt}=\frac{d(3t^{2})}{dt}+\frac{d(5t^{3})}{dt}-\frac{d(2t)}{dt}\).
For \(\frac{d(3t^{2})}{dt}\), using the power rule \(n = 2,a=3\), we get \(2\times3t^{2-1}=6t\).
For \(\frac{d(5t^{3})}{dt}\), with \(n = 3,a = 5\), we have \(3\times5t^{3 - 1}=15t^{2}\).
For \(\frac{d(2t)}{dt}\), using \(n = 1,a = 2\), we obtain \(1\times2t^{1-1}=2\).
So, \(v=\frac{dx}{dt}=15t^{2}+6t - 2\).

Step3: Substitute \(t = 2\) into the velocity formula

When \(t = 2\), \(v=15\times(2)^{2}+6\times2-2\).
First, calculate \(15\times(2)^{2}=15\times4 = 60\).
Then, \(6\times2=12\).
Now, \(v=60 + 12-2\).
\(v=70\space m/s\).

Answer:

\(70\space m/s\)