Question

The velocity of a sky diver t seconds after jumping is given by v(t) = 80(1 − e^(−0.2t)). After how many seconds is the velocity 60 ft/s? sec:

Answer 1

The equation representing the velocity is given as:

`v(t)=80(1-e^(-0.2t))`

It is required to find after how many seconds the velocity will reach 60 ft/s.

To do this, substitute v(t)=60 into the equation and solve for t:

`\begin{gathered} 60=80(1-e^(-0.2t)) \\ \text{ Swap the sides of the equation:} \\ \Rightarrow80(1-e^(-0.2t))=60 \\ \text{ Divide both sides of the equation by }80: \\ \Rightarrow(80(1-e^(-0.2t)))/(80)=(60)/(80) \\ \Rightarrow1-e^(-0.2t)=(60)/(80) \\ \text{ Subtract }1\text{ from both sides:} \\ \Rightarrow-e^(-0.2t)=(60)/(80)-1 \\ \Rightarrow-e^(-0.2t)=-(1)/(4) \\ \text{ Divide both sides by }-1: \\ \Rightarrow e^(-0.2t)=(1)/(4) \\ \text{ Take logarithm of both sides:} \\ \Rightarrow\ln(e^(-0.2t))=\ln(1)/(4) \\ \Rightarrow-0.2t=\ln(1)/(4) \end{gathered}`
`\begin{gathered} Divide\text{ both sides by }-0.2: \\ \Rightarrow(-0.2t)/(-0.2)=(\ln(1)/(4))/(-0.2) \\ \Rightarrow t\approx6.93\text{ seconds} \end{gathered}`

The time is about 6.93 seconds.