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

The time is about 6.93 seconds.