We are given the height function for a skydiver:

$h(t) = 10850 - 170t - 850e^{-0.2t}$

where $h(t)$ represents the height in feet and $t$ is time in seconds. To determine the time when the skydiver is falling at 120 feet per second, we need to find the velocity function, which is the derivative of the height function with respect to $t$.

Step 1: Find the velocity function

The velocity $v(t)$ is the derivative of $h(t)$, denoted as:

$v(t) = \frac{dh}{dt}$

We differentiate each term of $h(t)$:

1. $\frac{d}{dt}[10850] = 0$
2. $\frac{d}{dt}[-170t] = -170$
3. $\frac{d}{dt}[-850e^{-0.2t}] = -850 \times \frac{d}{dt}[e^{-0.2t}] = -850 \times (-0.2)e^{-0.2t} = 170e^{-0.2t}$

So the velocity function is:

$v(t) = -170 + 170e^{-0.2t}$

Step 2: Set velocity equal to -120 feet per second

We are told the skydiver is falling at a speed of 120 feet per second, which means $v(t) = -120$. Setting the velocity equation equal to -120:

$-170 + 170e^{-0.2t} = -120$

Step 3: Solve for $t$

Rearrange the equation to isolate the exponential term:

$170e^{-0.2t} = -120 + 170$ $170e^{-0.2t} = 50$ $e^{-0.2t} = \frac{50}{170} = \frac{5}{17}$

Take the natural logarithm (ln) of both sides to solve for $t$:

$-0.2t = \ln\left(\frac{5}{17}\right)$ $t = \frac{\ln\left(\frac{5}{17}\right)}{-0.2}$

Step 4: Compute the value of $t$

Now we will calculate the exact value of $t$.The time at which the skydiver is falling at a speed of 120 feet per second is approximately $t = 6.12$ seconds.

Would you like more details on the calculation or have any further questions?

Here are some related questions:

1. How does the skydiver's velocity change as time progresses?
2. What is the skydiver's velocity after 10 seconds?
3. Can we find when the skydiver reaches terminal velocity?
4. How would air resistance affect the height equation?
5. What is the skydiver's height after 6.12 seconds?

Tip: Derivatives help determine rates of change, such as velocity from a position function.