Main content

## SAT

### Course: SAT > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Quadratic and exponential word problems — Harder example

Watch Sal work through a harder Quadratic and exponential word problem.

## Want to join the conversation?

- sal am totally confuse even by rewatching it over and over.i rather skip this kind of question in SAT(234 votes)
- you can get the the decrease in the amount of subscribers for each 10 cent increase (for each choice ) then multiply the remaining amount with the the its selling cost.

For example,

10.65$ has a 13.5 cents increase from the original 9.30$ ... 13.5 x 20 = 270 ( the decrease in amount of subscribers) 2400-270 = 2130 ( the remaining amount of subscribers after the increase) 10.65$ x 2130 = 22,684.5 $ ( the revenue)

If you solved all the choices with the same technique you will find that

A) 1.35$ x 2400 = 3240 $

B) 9.30$ x 2400 = 22,320 $

C) 10.65$ x 2130 = 22,684.5 $

D) 22.80$ x -300 = -6840 $ ( not only will the company not gain any revenue ,but also they will lose 6840$ )(94 votes)

- This question can be answered using common sense by simply looking at the choices :

The first choice gives the option of $ 1.35 which will be too less as the costs won't be recovered.

The second choice gives the option of $9.30 which is no change and the question says the prices can be increased.

The third choice gives the option of $ 10.65 which is a fair increase.

And, the fourth choice gives the option of $ 22.80 which is extremely high and there won't be any more subscribers.(120 votes)- Sometimes math can be surprising, it's best not to take chances unless your under extreme time pressure.(34 votes)

- don't we only get like 50 minutes...(85 votes)
- he did this incredibly slowly. A better equation would be R(x) = (9.3 + 0.1x)(2400-20x), where R is equal to price (9.3 + 0.1x) multiplied by subscribers (2400-20x) and x is equal to the number of $0.1 increases to the price. This can be quickly punched into a calculator where the vertex can be found, multiplied by 0.1 and then added to the original price.(14 votes)

- after rewatching this video 3 times, i decided to skip this question(54 votes)
- its actually super easy , essentially what you do is use the options itself to get the answer.. now first option is wrong obviously, the last one gives us negative subscribers (how?, well subtract 9.3 from it and divide it by 0.1 and then multiply it by 20 (why did we do that? because if we subtract we get the amount that has been increased and once we divide with 0.1 we get how much it has been increased wrt 0.1 so we can directly multiply with 20 to see how many subscribers we have gained or lost)) if we repeat this process for 2 and 3 we see there is a slight profit for the third option.. YES, u can skip it and attempt it in the last if u want 800(3 votes)

- I am really confused about this, on the real SAT does it actually format like this? The most difficult thing is interpreting the word problems into function, is there a easier way to do this?(30 votes)
- Anybody else use the derivative of the function & set it equal to zero? The Calculus method is much more simple if you asked me..(18 votes)
- Yes, I did the same by using maxima(1 vote)

- Why does he divide by .10(11 votes)
- same thing i was thinking. He's supposed to multiply by 0.10 not divide. I'm so confused.(2 votes)

- At2:42, how is 20 divided by .10? In order to cancel out .10, wouldn't you multiply by .10, and get 2 instead of 200?(6 votes)
- To cancel out .10, you need to multiply 20 by the reciprocal of .10, which is 10. Its easier to see this with fractions. .10=1/10, so the reciprocal is 10/1.(4 votes)

- the easy way to think is as you can see you can cut out choice one and two also choice 4 (or you will have -subscription lol )

dont think too hard you know, time is running fast(6 votes)- you won't have negative subscription lol, you'd get an INCREASE in subs, but thing is that it's nowehere mentioned if they gain 20 sub per reduction of 10 cents.(1 vote)

- I have no idea what he is talking about. So confusing, Please can you upload an alternate form of this which is simplified.(5 votes)

## Video transcript

- [Instructor] Currently,
a local newspaper company sells print subscriptions
for $9.30 a month and has 2400 subscribers. Based on a survey conducted,
they expected to lose 20 subscribers for each $0.10 increase from the current monthly
subscription price. That's interesting. What should the newspaper company charge for a monthly subscription in
order to maximize the income, in order to maximize the income from the print newspaper subscriptions? So let's think a little
bit about their income. It's actually their revenue, how much money they're bringing in, but I get what they're talking about. So their income, let's
just say I for income, is going to be equal to the
number of subscribers they have, so S for subscribers, times
the price per subscriber, times the price per subscriber. So that's going to be their income. Now in this little, in the problem, they tell us that the
subscribers themselves are going to be a function of
price, and that makes sense. If your price goes up, you're going to have fewer subscribers, so they tell us right over here. Based on a survey conducted,
they expect to lose 20 subscribers for each $0.10 increase from the current monthly
subscription price. And the current monthly
subscription price is $9.30. So let's see if we can write our number of subscribers
as a function of price. So at the current price,
we have 2400 subscribers. We have 2400 subscribers. But we're going to lose 20 subscribers, so lose, so I'm gonna
subtract 20 subscribers for every, for every dime above $9.30. So let's multiply that,
so if we just take our, if we take P minus
$9.30, this would give us the absolute price increase,
how much we've gone above $9.30, and if we want to figure out how many dimes we are above $9.30, we can just divide that by $0.10. So this part right over
here, this tells us how many dimes above
our current price is P, and then for every one of those dimes, we're going to lose 20 subscribers. So this is probably the most interesting part of this problem. This is kind of the crux of it. How do you set up, how
do you set up subscribers as a function of price? Because once you do this,
you can then substitute back in here and then you're
going to have income purely as a function of price. Let me show you what I'm talking about. So before I actually even do that, let me simplify this a little bit. So subscribers is going
to be equal to 2400, and then, let's see, 20 divided by 0.1 is going to be 200 minus
200 times P minus $9.30. And then we can distribute this 200. So subscribers is going
to be equal to 2400 minus 200 P, minus 200 P, and then negative 200
times negative $9.30, let's see, that's going
to be 2 times $9.30, that's $18.60, but it's
going to be $1800 and, instead of $18.60, it's going to be $1860. So plus 1860, that's negative
200 times negative 9.3. And now we can add, we
can add this to that, and we are going to
get, we're going to get the subscribers are going to
be equal to 2400 plus 1860. If you had 1000, you'd get to 3400 and then you get 860, you get 240, 260. So you get 4,260 minus 200 P. So we now have a simplified subscribers as a function of price, and now we can substitute
this expression in for S, in for S in our original equation. So if we do that, we get income is equal to, instead of
writing S, we can write, we can write 4260 minus 200 P and then times P, times P. Now we can distribute,
we can distribute this P and we're gonna get income
as a function of price, is 4260 P minus 200 P squared. So income is a function of
price, it is a quadratic, and I actually like to write
my highest degree terms first, so I'll just, income is equal to, I'm just gonna swap these,
negative 200 P squared plus 42, 4260 P. So this is a quadratic, and it is a downward opening quadratic. We know that because the coefficient on the second degree term is negative. So this graph, I as a function of P, so this is, if this, so if this is the I
axis, that's the I axis, this is the P axis right over here, we know that it's gonna be, so
this is gonna be the P axis, we know it's gonna be a
downward opening parabola and that's good because we
want to find a maximum point. A downward opening
parabola will actually have a maximum point. If it was upward opening,
you'd be able to find a minimum point, but then there's no, it wouldn't be bounded onto the up side. So we need to figure out, we
need to figure out what price gets us to this maximum point, and this price, this is
going to be the vertex, this is going to be the P
coordinate of the vertex of this parabola. And let's just remind
ourselves how we can find the vertex of a parabola. There's multiple ways to do it. You can find the roots. You can find the x-values that
make this function equal zero and then the vertex is going to be halfway between those points. That's one way to do it. Another way to do it
is you could say, look, if I have something of the
form I is equal to a P squared plus b P plus C, instead
of writing y and x, I wrote I and P, well,
the vertex is going to be the coefficient, the
negative of this coefficient, negative b over two times a. That's going to be the P
coordinate of the vertex. P equals that is going to be when you hit the minimum or maximum point. In this case, it's going
to be the maximum point. So what's negative b? B is this right over here,
so negative b is 4260. That's gonna be over 2 a,
over 2 times negative 200, so that's gonna be negative 400. And so, what is that going to be? Negative divided by
negative is a positive. This is going to be the same
thing as 426 divided by 40. And let's see, can I simplify this? Well, let me just see, this is going to be 40 goes into 426 10
times with 26 left over. So it's going to be 10 and 26/40, or we can write this as, this
is going to be equal to 10 and 13, 13, 13/20. And so, if we wanted to write
that in terms of dollars because we're talking about a price, we would want to write this
in terms of hundredths, so this would be the same thing as 10 and, let's see, if you multiply
the denominator by five, you multiply the numerator
by five, 10 and 65/100, which is the same thing as $10.65. And luckily, we see that choice, we see that choice right over there. Another way we could have done it is we could have figured
out, what are the P values that it gets to zero income, and then the one that's
halfway in between those two is going to be where we hit our maximum. Halfway between your two roots are when you hit the maximum point, so that's another way that
you could actually tackle that and you could do that just by
looking at this original one. You can say, when does
this thing equal zero? So you can say, when does 4260 P minus 200 P squared equal zero? And since you have no constant term, you actually can just factor a P, so you say, P times 4260
minus P is equal to zero. So this is going to be equal to zero either when P equals zero,
and that makes sense. You're gonna have zero income
if you don't charge anything. Or if 4260 minus P is equal to. Oh, I'm going to be careful here. 4260 minus 200 P, minus
200 P is equal to zero. So that's going to be
equal to zero either when P is equal to zero or, or when 4260, 4260 minus
200 P is equal to zero. This is when you don't charge anything, you're obviously going to have no income. And this is when you charge so much, you're gonna lose all your subscribers and you'll have no income. Let's see, if you want to solve for, if you add 200 P to both
sides, you get 200 P is equal to 4260, divide
both sides by 200, you get P is equal to 4260 over 200. That's this price over here. That's the price at which you
lose all your subscribers. So your maximum point is going to be halfway in between these two,
halfway between zero and this, halfway between zero and that. Well, halfway between zero and that is just going to be half of this. So your maximum point is
going to be at 4260 over 400, which is exactly what
we figured out before. That's $10.65.