Math Model

[mathdad]

The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x = -(x/6)p + 100. Find a model that represents the revenue R as a function of x.
Note: Revenue = price x quantity sold or in short R = px.

Same as the previous problems.
Solve x = -(x/6)p + 100 for p and then plug into R = px.

x = -(x/6)p + 100

x - 100 = -(x/6)p

(x - 100)/-(x/6) = p

(6x - 600)/(-x) = p

R = [(6x - 600)/(-x)]x

Is it ok to leave this expression just as it is?
I managed to express the revenue R as a function of x.

 

[tkhunny]

(x - 100)/-(x/6) = p

Terrible notation. You fixed it in the next step. Good work.

R = [(6x - 600)/(-x)]x

Sometimes, it is instructive to leave such things on different forms so that we can learn something. When just manipulating expressions, it seems prudent to simplify if and when you can.

In this case: R = 600 - 6x = 6(100-x) are quite a bit simpler than where you have left it.

 

[Otis]

... Solve x = -(x/6)p + 100 for p and then plug into R = px ...

R = [(6x - 600)/(-x)]x

Is it ok to leave this expression just as it is?

I would simplify it; good practice for you, too. I see an x/x situation, and that's easy to simplify.

?

 

[Steven G]

It is best that the denominator not have a negative sign, it is just mathematical grammar.
This is why we learned things like a positive divided by a negative is a negative.
I would certainly take off a point if a student left their final answer as 7/-2 instead of -7/2

 

[mathdad]

It is best that the denominator not have a negative sign, it is just mathematical grammar.
This is why we learned things like a positive divided by a negative is a negative.
I would certainly take off a point if a student left their final answer as 7/-2 instead of -7/2

I made small errors. It's ok. I am not preparing for a test. This is a review of material learned long ago.

 

[mathdad]

I would simplify it; good practice for you, too. I see an x/x situation, and that's easy to simplify.

?

I now know to simplify as much as possible.

 

[mathdad]

Terrible notation. You fixed it in the next step. Good work.



Sometimes, it is instructive to leave such things on different forms so that we can learn something. When just manipulating expressions, it seems prudent to simplify if and when you can.

In this case: R = 600 - 6x = 6(100-x) are quite a bit simpler than where you have left it.

I could have simplified a bit further but considering that I am not a classroom student preparing for a test to gain credits, I guess my laziness can be forgiven.

 

[Otis]

I now know to simplify as much as possible.

I think that's a good idea, to refresh your memory by practicing. Eventually, you'll recognize simplifications without needing paper and pencil, and you can then decide whether you want to write them out. Practice also helps improve symbolic-reasoning skills, to work with abstraction in later topics.

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Here's an alternate substitution, for expressing revenue as a function of quantity sold.

... x = -(x/6)p + 100

... R = px

When I examine the right-hand side of the first equation, I see 1/6th of xp being subtracted from 100. And xp is px, which we know is R.

x = 100 - (xp)/6

x = 100 - R/6

-6x = -600 + R

600 - 6x = R

?

 

[mathdad]

I think that's a good idea, to refresh your memory by practicing. Eventually, you'll recognize simplifications without needing paper and pencil, and you can then decide whether you want to write them out. Practice also helps improve symbolic-reasoning skills, to work with abstraction in later topics.

---

Here's an alternate substitution, for expressing revenue as a function of quantity sold.


When I examine the right-hand side of the first equation, I see 1/6th of xp being subtracted from 100. And xp is px, which we know is R.

x = 100 - (xp)/6

x = 100 - R/6

-6x = -600 + R

600 - 6x = R

?

Is it interesting to know that we can switch expressions when the first of two has a negative. Say we have -a + b. I can express this to be b - a, which is what you did above concerning 100 in that model.

 

[Otis]

Is it interesting to know that we can switch expressions when the first of two has a negative ...

Yes. We can switch the order, and the signs can be anything. What matters is the operation (addition).

There's a general property for addition (for multiplication, too) called the 'commutative property'. When we add two numbers (or multiply them), we may write the numbers in either order.

x + p = p + x

xp = px

Also, once students understand negative numbers, they can view subtraction as a form of addition. That is, subtracting any number is the same as adding its opposite.

b - a = b + (-a)

Well, if we view the expression b-a in terms of addition, then we're free to use the commutative property of addition, to change the order of the numbers being added.

b - a

b + (-a)

-a + b

The expressions below are equal.

-(a - b)

-a + b

-(-b + a)

b - a

?