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### Course: Algebra (all content) > Unit 7

Lesson 18: Modeling situations by combining and composing functions (Algebra 2 level)# Modeling with function combination

Sal models the height of a tree on top of a tower, by adding the functions that model the growth of the tree and the tower separately.

## Want to join the conversation?

- I don't know when your suppose to multiply them or add them or subtract them or divide them. I keep getting all of the practice questions wrong and its honestly frustrating me. could you please help me. when do I do which?(38 votes)
- I completely agree and this video does not help with ANY of the quiz questions.(23 votes)

- There should be more videos that explain how to do this.

Also, when I was doing the activity for this video, the question about Haruka's ponch factory doesn't make sense. Can you explain what the prompt is saying?(20 votes)- In order for a company to determine the total profit(the amount of money it gets to keep after expenses are covered), it must subtract its total expenses(costs in the form of wages, land taxes, production costs,etc.) from its revenue(money made from the distribution of a product/service). Thus, Profit=Revenue-Costs.

In this example, the total revenue depends on both how many ponchos are sold and the sales price of each poncho. Since each poncho sells for $18, the money made from the sale of one poncho is 18; the amount made from the sale of two ponchos is 2*18=36; the amount made from the sale of three ponchos is 3*18;; and so on. So in general, the total revenue for selling n ponchos is 18*n=18n.

The problem describes the total profit as a linear function of n(the number of ponchos sold). I say "linear" because for every additional poncho sold, the profit increases by $17, starting with a profit of $12 for one poncho sold. This means that for n=1, P(1)=12, P(2)=12+17=29, P(3)=17+29=46, .....and so on. Thus, profit function is best modeled by the equation P(n)=17(n-1)+12=17n-17+12=17n-5. You could derive this using the point slope formula of a line.

With both the revenue and profit functions, you should be able to solve for the cost function on your own using P(n)=R(n)-C(n).(18 votes)

- Is there another video about this? A much more clear one(14 votes)
- I'm sure there are better resources on youtube about modeling with function combination.(3 votes)

- In the modeling with combined functions practice problems, when the problem states "by a factor of" something, what does that mean mathematically?(3 votes)
- "by a factor of" means multiplication. It can also mean division if something is
*reduced*"by a factor of." Look out for when it's combined with the phrase "for each," though, because then you're looking at exponents. Here are some examples.

My electricity usage increases by a factor of four in the summertime. In the winter, I spend $15 / month on electricity. So in the summer months, I spend $60 / month. 15 x 4 = 60.

By bringing my lunch to work, I have reduced my spending on food by a factor of six. I used to spend $240 / month on lunch. Now I spend only $40 / month on lunch. 240 / 6 = 40.*Or*, 240 x 1/6 = 40.

Every time I drop a cookie on the floor, the number of ants in my house increases.*For each*cookie I drop, the number of ants increases by a factor of three. I started with six ants, and then I dropped two cookies. Now there are 54 ants. 6 x 3 x 3 = 54, or 6 x 3² = 54.

To look at that last one another way, since it's a little confusing, you can break it apart to consider the cookies separately. I had six ants, and I dropped a cookie. 6 x 3 = 18. Now I have 18 ants, and I dropped another cookie. 18 x 3 = 54.(10 votes)

- Okay this is 4th question for this problem and its baffles me. here it is.

The number of students, S, serviced by the school system in the town of Emor, t years from 2000 can be modeled by the function S(t) = 10,000(1.1)^t. The number of classrooms, C, in the town of Emor, t years from 2000 can be modeled by the function C(t)=450 + 40t.

Let D be the average number of students per classroom in Emor's school system t years from 2000.

Write a formula for D(t) in terms of S(t) and C(t).

D(t) =?

Write a formula for D(t) in terms of t.

D(t) =

My answer from D(t) in terms of S(t) and C(t) is = [2000 + S(t)]/[2000 + C(t)]

then D(t) in terms of t is =[2000 + 10,000(1.1)^t]/[2000 + 450 + 40t]

= [2000 + 10,000(1.1)^t]/[2450 + 40t]

Please verify my answer if I'm correct.. Please provide correct answer if I'm wrong. I made the last 3 questions correct, and don't want to fail this last question which really makes me stump..(4 votes)- "t" is always defined as the number of year from 2000. So, you don't need to add the 2000's into your function.

The average students / classroom = S(t)/C(t)

D(t) = [10,000(1.1)^t] / (450 + 40t)

Hope this helps.(7 votes)

- this problem is dealing with addition but how do you deal with multiplication or division?

also how do you know when to add, subtract, multiply, or divide

thanks in advance(6 votes)- Hey, so one of the questions in the exercise has something along these lines: C(t)=400t+30 is the number of batches of corn made year t, H(t)=30t+15 is the price per batch in year t, B(t) is the total profit in year t, what is B(t)?

well, you have C(t) batches and you get H(t) per batch, so B(t)= C(t) x H(t),

then to find B(t) in terms of t, perform the multiplication:

B(t) = (400t + 30)(30t + 15)

-Remember the FOIL method for multiplying out brackets,

F-first, 400t x 30t

O-outside, 400t x 15,

I-inside 30 x 30t,

L-last, 30 x 15

B(t)= 12000(t^2) + 6000t + 900t + 450

B(t) = 12000(t^2) + 6900t + 450,

which is the function for how much profit you will make!

Hope this helps a bit:):):)

If you find a question you need help with, just let me know!(2 votes)

- How do you know when to add, subtract, multiply or divide for the equation?(5 votes)
- Modelling with function combination

One college states that the number of men, M, and the number of women, W, receiving bachelor degrees t years since 1980 can be modeled by the functions

M(t) = 526 - t and W(t) = 474 + 2t

Let N be the total number of students receiving bachelor's degrees at that college t years since 1980

Write a formula for N(t) in terms of M(t) and W(t).

N(t) =

Write a formula for N(t) in terms of t.

N(t) =

My answer for the 1st is N(t) = M(t) + W(t)

For 2nd question:

N(t) = 526 - t + 474 + 2t = 1000 + t answer//

Can anyone verify my answer..Please provide correction if I'm wrong.(4 votes) - hi,i just have one question:

are there any more videos on this topic?

im having alot of problems doing the practice exercises

could someone please help me. ill greatly appreciate any help

thanks alot(5 votes)- Did you figure it out yet? It’ll be grateful for some help here.(1 vote)

- What are the limitations of mathematical modeling?(5 votes)

## Video transcript

- [Voiceover] Ify is
building a tree tower, which is a tower built on top of a tree. The tree is currently five meters tall and Ify has found, I don't
know if it's Ify or Ify, and Ify has found that it is growing by zero point one meters a month or a tenth of a meter a month. The tower is currently two meters tall. So this tower that sits
on top of the tree, is two meters tall
currently and Ify, or Ify, adds to it at about zero
point two meters a month. So, I guess, he or she, I don't know, Ify, let's just say, "he." Is continuously building this tower on this continuously growing
tree, which is fascinating. Alright, the function A
of m, returns the tree's height in meters and months from now. Fascinating. The function B of m,
returns the tower's height in meters and months from now. So, this is the tree's height,
A of m, is the tree's height. B of m is the tower's height. Find the formula of the two functions. So, A of m, so they tell us the tree is currently five meters tall. So, it's going to be five
meters tall right at the start. And then every month it is
growing by zero point one meters, so it's going to be five
plus zero point one times m. And this m here, this is not meters, this is actually the months. Remember, m returns, m
is the number of months. So after zero months,which is right now, well this is just going to be five. After one month, it's
going to be five point one. After two months it's
going to be five point two. Which is exactly what we want. Alright, now let's think about the tower. So, the formula for B of m, so the tower is currently two meters tall. So, it's currently two meters tall and it grows at two tenths
of a meter per month. So, two tenths times the number of months. And once again, this m right
over here is not meters, I'm not writing the units here, we're just assuming whatever
this returns is in meters. This m right over here is the
number of months that pass by, the number of months from now. Alright, the function C of m
returns the vertical distance between the ground and
the top end of the tower. Makes sense, that would be
from the bottom of the tree to the top of the tower. Wright the formula, C of m,
in terms of A of m and B of m. Well the total height is going
to be the height of the tree, which is A of m plus
the height of the tower. Plus B of m, plus B of m. That's what C of m is going to be. And then they say, "Wright the formula of
C of m in terms of m." Well, we just need to
add these two functions. So, if we add five plus zero point one m to two plus zero point two m, that's going to be, we
could add five plus two, and we'er going to get seven, plus, and if I have point one m
and I add another point 2 m, that's going to be point 3 m. Zero point 3 m, and we are done. And we got it right.