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## Precalculus

### Unit 1: Lesson 2

Modeling with composite functions

# Modeling with composite functions: skydiving

Sal models the maximum speed of a skydiver, by using composing given formulas for the maximum speed as a function of parachute area and parachute area as a function of width. Created by Sal Khan.

## Video transcript

- [Instructor] We're told that Flox is a skydiver on the planet Lernon. The function. A of w is equal to 0.2 times W squared, gives the area, A, in square meters under Flox's parachute when it has a width of W meters. That makes sense. The function V of A is equal to the square root of 980 over A gives Flox's maximum speed in meters per second when she skydives with an area of A square meters under her parachute. All right. Write an expression to model Flox's terminal velocity when her parachute is W meters wide, and then they want us to evaluate the terminal velocity when her parachute is 14 meters wide. Well, let's just focus on the first part first. Pause the video and see if you can have a go at that. All right. Now let's just think about what they're asking us. They want us to model terminal velocity when her parachute is W meters wide. So really what they want us to do is come up with a terminal velocity. Let's call that V, that is a function of W, that is a function of the width of her parachute. Well, we have a function here that gives terminal velocity as a function of the area of her parachute, but lucky for us, we have another function that gives us area as a function of width. And so we could say this is going to be the same thing as V of this function, right over here. I'll do it in another color, A of w. And so that is going to be equal to, let me keep the colors consistent. Well, everywhere, where I see an A in this expression, I would replace it with A of W, which is 0.2 W squared. So it's going to be equal to the square root of 980 over instead of A, I am going to write, instead of this, I am going to write 0.2 W squared, because that is A as a function of W. 0.2 W squared. So this right over here, this is an expression that models Flox's terminal velocity, V, as a function of the width of her parachute. So that's what we have right over there. And then the next part, they say, what is Flox's terminal velocity when her parachute is 14 meters wide? Well, then we just have to say, okay, W is 14. Let's just evaluate this expression. So we'll get the square root of 980 over 0.2 times 14 squared. Well, 14 squared is 196 and this would be equal to the square root of, see, 980 divided by 196, I believe is exactly five. So this would be five divided by 0.2. And so, five divided by essentially one fifth is the same thing as five times five. So this would be the square root of 25, which is equal to five. And the terminal velocity, since we gave the width in meters, this is going to give us the maximum speed in meters per second. So five meters per second, and we're done.