The velocity of a sky diver t seconds after jumping is given by $v ( t ) = 80 \left( 1 - e ^ { - 02 t } \right)$ . After how many seconds is the velocity 50 ft/s?
A)5.0 s
B)4.9 s
C)8.9 s
D)0.2 s
E)none of these

Question

Quizplus · Accepted Answer

To find when the velocity is 50 ft/s, we set $v(t) = 50$ and solve for t:

$$50 = 80 (1 - e^{-0.2t})$$

Dividing both sides by 80, we get:

$$0.625 = 1 - e^{-0.2t}$$

Subtracting 1 from both sides gives:

$$-0.375 = -e^{-0.2t}$$

Taking the natural logarithm of both sides (ln is the natural logarithm function), we have:

$$\ln 0.375 = -0.2t$$

Solving for t, we get:

$$t = \frac{\ln 0.375}{-0.2} \approx 4.9\text{ s}$$

Therefore, the velocity is 50 ft/s after about 4.9 seconds.

The Velocity of a Sky Diver T Seconds After Jumping v(t)=80(1−e−02t)v ( t ) = 80 \left( 1 - e ^ { - 02 t } \right)v(t)=80(1−e−02t)

The Velocity of a Sky Diver T Seconds After Jumping $v ( t ) = 80 \left( 1 - e ^ { - 02 t } \right)$