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### Course: 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.

## Want to join the conversation?

- What does terminal velocity mean?(2 votes)
- The faster you move through the air, the greater the force the air exerts on you. So if you're in freefall, you speed up until the force of the air pushing you up matches the force of gravity pulling you down. Once this happens, the net force on your body is 0, so your speed remains constant from then on (unless you change your position, the air thickens, or you hit the ground).

The final, constant speed that you reach is called your terminal velocity.(27 votes)

- Like many subjects on Khan, I understood the video well enough but am struggling with the exercises. I have no problem figuring out what the input and output for a mediating function needs to be, but I'm having trouble putting both functions together and composing the actual equations.

Here's a problem I bombed (shortened):

When a lecture video is 10 minutes long, 100 students watch. Every minute over, the number of students decreases by 3.

V(x) returns the average student grade based on the number of students that watched. G(k) returns the average grade based on the length of the lecture. Write G(k) in terms of V and x. I knew the mediating function M(y) had to return how many students watched the lecture given the length. Where I was stuck was the equation itself. The answer was M(y)= 100-3(y-10). Unfortunately I did not write down G(k) at the time and so do not remember the final answer. My question is: how can you compose the equation M(y)=100-3(y-10) given the word problem? It makes sense, but I seem to be unable to take information and make an equation out of it. Does anyone have tips for composing equations? I thought I might need to backtrack in math a little and review, but I'm not sure at this point.

Sorry this was so long! Thanks.(6 votes)- In your question you put M(y), but it was actually M(k), meaning K as in minutes, or at least it was when i did it.. With that said i'm going to say this question also made no sense to me.

M(k)=100-3(k-10) seems weird. K is supposed to be minutes, so what if the video was two minutes long?

M(k)=100-3(2-10)

M(k)=100+24

M(k)=124

So this function says that if the video lasts two minutes, then it's watched by 124 students? The only way that function makes sense is if you put in something over positive ten. So they're basically saying that K must be greater than ten, but nowhere in the function was that clarified. I mean that was the whole difficulty of the problem. "How can i write a function that will -3 from 100 when the video is over ten minutes.?" And it doesn't look like they showed how. Maybe i misread something. I don't know.(2 votes)

- But didn't the question ask us to round our answer to the nearest 10?(5 votes)
- He said that just so as not to give any clue to what kind of answer works.(0 votes)

- At2:58When solving the problem Why did he divide 980 by 196 before he multiplied 196 by 0.2?(3 votes)
- Either way will work. It's your choice. Sal chose to reduce the fraction by one factor at a time. He could have multiplied and then reduced the fraction.(5 votes)

- why is it V(d(t)) ...arent we trying to figure out v(t)?(3 votes)
- This notation leads to what appears to be conflicting information. The original function is given as V(d) = rule involving d. The end result of the composition in this video is V(t)=a different rule involving t. Shouldn't the end result be noted as V composed with d at t as the new function that results from the composition instead of reusing the function name V?(3 votes)

- At2:17, when Sal squares the expression in the brackets, how come you don't have to do it in the same way that you would when dealing with the square of a polynomial expression?(3 votes)
- Well, the first term being squared is just a number. 1/sqrt3 is a number without any variable in it. The only variable present is under sqrt(t-5) such that this can be considered a lone expression. If you were to solve (2x)^2, you would square the number first, and then the variable separately, right? So the same principle applies here.(4 votes)

- I think the expression can be simplified to 70/w, can it?(4 votes)
- Which expression? They don't look like they can be simplified to 70/w though. They're supposed to be pre-simplified.(1 vote)

- 3:50Why is it that we are able to evaluate V(2)? Seems like the domain of the composition should be x> or = 5. Thanks!(2 votes)
- Good question.

After you compute V(d(t)), then simplify, you get 4t^3, and sqrt(t-5) is gone. Does that mean it no longer matters? I think it might - I'm not sure though.(2 votes)

- My problem with this section is that I don't understand how to put the functions together from the word problems! The video makes everything clear: ok, yeah, I can do that. And then the practice problems are so different that I'm completely thrown off.(3 votes)
- is this concept really hard or am I that stupid?(2 votes)
- You definitely aren't alone in this. For example, I had problems with rates and ratios, but composing functions is really easy for me. That's because I have a lot of experience with it in functional programming.(2 votes)

## 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.