So in this video I just

want to talk about a couple of application

problems, or maybe just one application problem

for compositions. So the first problem, or the

problem that I want to look at, in the video is from Calculus

for Business, Economics, and the Social

Sciences by Hoffmann and several other authors. It’s the 11th edition. And the problem goes

something like this. Arthur, the manager of

a furniture factory, finds that the cost of

producing q bookcases– so this is q– during the

morning production run is C of q is equal to q

squared plus q plus $500. So we have a cost function

based on a quantity. On a typical workday,

q of t– that is the quantity as a function

of time– equals 25t bookcases. And they are produced

during the first t hours of a production run

for 0 less than or equal to t is less than or equal to 5. We have a little

statement about the domain of this q of t function. So part a says, “Express the

production cost in terms of t.” So right now I have cost in

terms of quantity, right? I have these two functions. And the reality is,

I want to figure out what the cost is with

relation to time. So this is a composition

of functions, right? q of t gives me an

output that’s quantity, and I can put that output

into this cost function to figure out the

overall function. So I will find C of q of t,

put this 25t in everywhere there’s a q in C.

So let’s do that. So I have 25t squared

plus 25t plus 500. When I square 25t, this

square is on the 25 and the t. So 25 times 25 is 625. Distribute that squared. 25t plus 500. So this is the cost as

a function of time now. So now let’s go to

the next question. How much will have been

spent on production by the end of the third hour? So what’s the cost at the

end of the third hour? I’m going to rewrite this cost

function that we just found, 625t squared plus 25t plus 500. And I am trying

to find cost of 3 when t equals 3, the third hour. So I’ll plug in 3. It’s a time

measurement so I know that it goes into

the time variable. Let me plug that

into my calculator, and you guys do

the same, please. Check me. And I get that the

cost is $6,200. Now, what’s the third question? What’s the average cost of

production during the first 3 hours? And remember, I think we

talked about a problem with average cost is equal

to the cost function, C of whatever, divided

by the quantity. So it’s cost per unit. I’m trying to get an

average cost per each unit. So in this case my

average cost should be C of t divided

by q of t, because I have both of those

in terms of t. And maybe you’re thinking

of this as C of q of t, this composition

function like this. Either way, I want to do

it for the first 3 hours. So I’m really thinking of

C of 3 divided by q of 3. So from the previous problem

I know this one, right? I found the cost at the end of

the third hour to be $6,200. I want to find q of 3. Let’s do that. q of

t is equal to– what was it equal to again? 25t. 25 times 3 is equal to 75. It looks like I can make 75

bookcases in the first 3 hours. So I’ll plug those two numbers

in, 6,200 divided by 75. And I get an average

cost of $82 and it looks like $0.67 per

bookcase if the cost to produce the bookcases

is evenly divided. I think there’s one more

question in this problem. Arthur’s– this

should be budget. There’s a typo there–

allows no more than $11,000 for production during the

morning production run. When will this limit be reached? So I have one quantity

here, $11,000. And I have to figure out

where that number goes in all of the formulas

that I have so far, right? So I have a cost

function, C of q of t. Let me write that down again. That’s 625t squared

plus 25t plus 500. I also have a quantity function,

25t I think it is, right? And so I’m trying to figure

out where does this $11,000 go? q is a quantity of

bookcases that’s in units. t is in time, that

should be in hours. And cost is my only one

that’s in money, right? So I should plug in

this $11,000 right here into this side for

the cost and figure out when that is, right? When will this limit be reached? When is a question that’s

answered by an answer in t, in number of hours, right? That’s a time measurement. When is a time question. So I am going to go ahead

and plug in 11,000 here. And then I get this

quadratic formula or this quadratic equation. So I want to go ahead

and solve here for 0. I’ll subtract 11,000

from both sides. I think I get minus

10,500 on this side. And then 25 is

definitely factorable. You can factor it out

of here, if you want. When I do that, let me divide

10,500 divided by 25 to figure out– this should be

minus 420, this last one. So you can do that. That doesn’t look

like it helps me. I definitely don’t

want to– gosh, I would hate to try

and factor that thing. I’m just going to go ahead

and use the quadratic formula since I know what it is. I know that it’s going to work. And it looks like factoring

is going to be a pain. So let me go ahead and do that. So I have t is

equal to– and this is the negative b plus

or minus the square root b squared minus 4ac over 2a. So t is equal to negative– and

here, since I factor this out, I’m going to go ahead and

use these smaller numbers. Here’s my a is 25, b is positive

1, and c is negative 420. So I’m going to use

those numbers in here. So negative 1 plus or

minus the square root 1 squared minus 4 times 25 times

negative 420 over 2 times 25. Now, you can do these

in different parts, if you want to. I’m going to go ahead

and plug this whole thing into my calculator

to avoid rounding. So I use parentheses. And when I plug in– let’s see,

I’m going to do the plus 1. Negative 1 plus this is

the one that I’m doing. And there’s a 50 down here. If I do this– and I just

want to make this note– if I use this minus

version, negative 1 minus the square

root of whatever, I’ll get a negative

number, right? And you can try that

on your calculator. You get a negative

number, which isn’t really a good output in terms of time. I mean aside from

Back to the Future, we don’t really talk about

negative values of time. So we want to go ahead and

just use the positive value. I guess somehow I thought

the Back to the Future joke was actually funny, when it’s

really not but, you know. So I would go ahead and

put these numbers in. And when I do that, I get

t is equal to 4 point– I think I get a 08 when

I round it– hours. So right around 4 hours

I hit this $11,000. A little bit over 4 hours I

hit the $11,000 mark for cost. So that’s that problem. Let me see. Let’s take a look at

this next problem. It may be a couple questions. This is from a

different calculus book, Essential Calculus by

Wright, Hurd and New and published in 2008. And the problem looks like this. Two students create

a computer program which connects

the dots on a grid so that two players can

play Chase the Rabbit. They buy blank CD discs at a

local store for $0.50 and sell them for $5. They pay another student

$27 per day, 5 days a week, to answer the phone,

take orders, relay customer questions, and make

duplicates of the master disc. And it tells you, let x denote

the number of discs made. Let’s C of x be the weekly

cost function, the total cost, and let R of x be

the revenue function. So the first question says

write an expression for cost, for C of x. So we’ve gone through

an application problem. I encourage you guys, if

you printed this problem out to go ahead and try

it again on your own. Write this cost function. Part b is write the

revenue function. Go through and then come

back and compare and see what you get. So I want to write the

expression for C of x. I’m telling you, you

should push pause now. I know, spoiler alert. I’m about to go to

the problem, right? Write the expression

for C of x, and then what is the weekly cost of

producing 500 discs, so cost. So I have two

parts to the cost I think when I read

through this problem, and sometimes you might have

to read through it twice. But they’re buying

discs for $0.50 each, and they also have a pay out to

this other student of $27 per day. So the cost is the cost of

discs plus the student, right? That’s what they’re

spending money on. So the disc cost $0.50 each

is how I interpret that. So I get 0.50. And let’s say the

number of discs is x. I think that’s

what we said, let x denote the number of discs made. Plus what it costs to pay

the student, $27 a day times 5 days a week. Oops, I put a 5 in there. 27 times 5 days a week. So the cost should be 0.5x

plus– and I think this is 135. Now, the second

part of the question says, what is the weekly

cost of producing 500 discs? And this is what I would call

a C of x function, right? Cost as a function of x. So C of 500, how much does

it cost to produce 500 discs, 0.5 times 500 plus 135. So let me put that

into my calculator. And I get $385. Is that what you guys got? Good. And that seems reasonable to me. You’re making 500 discs. They each cost you $0.50. So that should be around

250 plus this other payout. Part b, write the

revenue function. Revenue– how much money

they’re bringing in, right? So here in this revenue

function we, I think, are bringing in

money from one place. And look back in your

problem, I believe they’re selling the

discs for $5 each. So I did my revenue

function– again, x is the number of discs– is

$5 times however many they sell. So I think that’s

kind of really simple, but I don’t think they’re

making money in any other way. Now, how much

revenue is produced by the sale of 500 discs? R of 500 equals 5 times 500. I think I get $2,500. They make $2,500 by

selling 500 discs. Let’s do part c. Write a profit function. Profit is how much you

bring in– revenue– minus how much you had to

put out– cost, right? So in this case, R

of x minus C of x. Our R of x is 5x and our cost

function is 0.5x plus 135. So let’s simplify this. 5x minus 0.5x minus 135,

that’s 4.5x minus 135, I think, for profit based on

the number of discs. And determine the profit

from the sale of 500 discs. Profit from 500

discs equals– this equals the revenue of 500,

which I’ve already found, minus the cost of 500. Or you could put 500

into the profit function. But I’ve already

plugged these numbers in and I know what they

are, so let me go ahead and put that in here. My revenue is 2,500

and my cost was 385. So let me– on the calculator

I get $2,115 per week, right? This is a weekly profit. So that’s pretty good. And I think that’s the

weekly profit based on sales of 500 discs. D, how many discs must be

sold in order to break even? This idea of breaking even

is kind of what it sounds. It’s the point where you’re

putting out as much money as you’re bringing in. So where R of x is

equal to C of x, this is called your

break even point. So I’m just going to do

that, plug in my functions. Where’s my function? R of x was 5x and the cost

function was 0.5x plus 135. And I’ll just solve for however

many discs I need to sell. So I subtract 0.5

from both sides and then divide by 4.5, 135. And I get x is equal to–

oops, not money, right? x is of quantity is 30 discs. The last one which I would

like you guys to think about, a business professor estimates

that the campus craze– I have a lot of typos

here– craze for the game could become national. And therefore, the game

could be marketed nationally. If 100 colleges were

to become market sites, how might the cost and

revenue functions be affected? Based on your thoughts,

write new functions for cost, revenue, and profit

for the 100 colleges combined together. So I’m not going to do this. Will you guys think about

this and talk about it with your fellow

classmates, how you think that this

would be affected? But that’s the end of

functions and applications of compositions. So let me know if

you have questions.