If an average-sized man with a parachute jumps from an airplane, he will fall 
F(t)=12.5(0.2^t−1)+20t
 feet in t seconds. How long will it take him to fall 150 feet?


We are given the function \( F(t) = 12.5(0.2^t - 1) + 20t \), which describes the distance an average-sized man falls with a parachute in \( t \) seconds. We need to find how long it will take him to fall 150 feet. This means solving for \( t \) when \( F(t) = 150 \).

### Step-by-Step Solution:

1. Set \( F(t) = 150 \):
   \[
   12.5(0.2^t - 1) + 20t = 150
   \]

2. Expand the equation:
   \[
   12.5 \times 0.2^t - 12.5 + 20t = 150
   \]

3. Move the constant term \(-12.5\) to the other side:
   \[
   12.5 \times 0.2^t + 20t = 162.5
   \]

This equation involves both an exponential and a linear term, which cannot be solved algebraically. Numerical methods such as Newton's method or a graphing approach would be needed to find \( t \).

I will now compute the approximate solution numerically.The approximate time it will take the man to fall 150 feet is \( t \approx 8.12 \) seconds.

Would you like more details or have any other questions?

Here are five related questions for further exploration:
1. How would the equation change if the man's parachute had a slower rate of descent?
2. How much distance does the man fall after 5 seconds?
3. What would the function \( F(t) \) look like if there were no parachute?
4. Can you find the total distance fallen after 10 seconds?
5. How would you find the time to fall 200 feet instead?

**Tip:** When solving equations with both exponential and linear terms, graphical methods or numerical solvers are often the most efficient.

If an average-sized man with a parachute jumps from an airplane, he will fall F(t)=12.5(0.2^t−1)+20t feet in t seconds. How long will it take him to fall 150 feet?

Explore how to solve a problem involving the fall of a man with a parachute using the function F(t) = 12.5(0.2^t - 1) + 20t. Learn how to apply numerical methods to find the time required to fall 150 feet. Suitable for high school students in Grades 10-12.

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