Composite functions to model extraterrestrial skydiving

We're told that Phlox is a skydiver on the planet Lernon. The function A of w is equal to 0.2 times w squared, which gives the area A in square meters under Flux'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 900 over 980 over A, which gives Flux'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 Flux's terminal velocity when her parachute is w meters wide. 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. So we could say this is going to be the same thing as V of this function right over here; I'll do another color: A of w.

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—so 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 Flux'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 Flux'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—let's 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 5 meters per second, and we're done.