Algebra word problem difficulty defining function

[late347]

Homework Statement

cellphones are sold at price of 302$ per unit
In one week there were 450 units sold.

for every one dollar increase in unit price, the weekly sales of the phones fell by four units.
form a function which describes the weekly sales income when the phone's price changes by X dollars
Can the store sell cellphones at the price of 522 dollars?

Homework Equations

3. The Attempt at a Solution [/B]

I know that sales = turnover. But it does not seem to be the way to properly calculate this problem.

I had the wrong function evidently as my teacher sent me the correct solution to this problem. But I wonder if my original way is sensible or not.

I wonder if I should have somehow created a function which describes the weekly sales, from a zero units sold beginning. Like at the beginning hours of the business week, obviously nothing has been sold at that beginning hour. Past weekly sales would not matter in this sense, because new weekly sale is being calculated.

Normally of course turnover is the units sold times the price/unit.
Immediately it looks like at some point. The unit sale amount will become negative, when the price increases too high. Worst case for store is that nothing is sold = 0 unit sale amount.


302 with no price change would equal 450 units
302+1 would make the change sales to 450 -4

304 would make sales to 450 - 4*2


turnover = y

y = (price/ unit) x (unit amount sold)

y = ( 302 + x) * (450- 4x)

Can the store sell phones, at the unit price of 522?

522- 302 = 220

(302+220) * (450-4*220) = y

522 * ( 450 - 880) = y
522 * - 430 = y

-224460 = y

looks like nothing will be sold at that price,
This looks like it will be the case when one looks at the right hand side inside the brackets ( 450 - 4*220). This number should have been a positive number, for any amount of unit sales to have occured.

 

[SteamKing]

Homework Statement

cellphones are sold at price of 302$ per unit
In one week there were 450 units sold.

for every one dollar increase in unit price, the weekly sales of the phones fell by four units.
form a function which describes the weekly sales income when the phone's price changes by X dollars
Can the store sell cellphones at the price of 522 dollars?

Homework Equations

3. The Attempt at a Solution [/B]

I know that sales = turnover. But it does not seem to be the way to properly calculate this problem.

I had the wrong function evidently as my teacher sent me the correct solution to this problem. But I wonder if my original way is sensible or not.

I wonder if I should have somehow created a function which describes the weekly sales, from a zero units sold beginning. Like at the beginning hours of the business week, obviously nothing has been sold at that beginning hour. Past weekly sales would not matter in this sense, because new weekly sale is being calculated.

Normally of course turnover is the units sold times the price/unit.
Immediately it looks like at some point. The unit sale amount will become negative, when the price increases too high. Worst case for store is that nothing is sold = 0 unit sale amount.


302 with no price change would equal 450 units
302+1 would make the change sales to 450 -4

304 would make sales to 450 - 4*2


turnover = y

y = (price/ unit) x (unit amount sold)

y = ( 302 + x) * (450- 4x)

Can the store sell phones, at the unit price of 522?

522- 302 = 220

(302+220) * (450-4*220) = y

522 * ( 450 - 880) = y
522 * - 430 = y

-224460 = y

looks like nothing will be sold at that price,
This looks like it will be the case when one looks at the right hand side inside the brackets ( 450 - 4*220). This number should have been a positive number, for any amount of unit sales to have occured.

Another way to look at this problem is to define x as the number of dollars difference in the price of a single phone, above or below the base price of $302.
You know that the change in weekly sales is 4 phones less than the base sales of 450 phones for each dollar increase in the price.

Therefore, Number_of_Sales = 450 - 4 ⋅ x

Now, if no phones are sold, you set Number_of_Sales = 0 in the formula above, and obviously sales income will equal zero.

You can calculate the amount of price increase x beyond which no phones will sell. All that's left after that is to see if 302 + x < 522.

 

[late347]

Another way to look at this problem is to define x as the number of dollars difference in the price of a single phone, above or below the base price of $302.
You know that the change in weekly sales is 4 phones less than the base sales of 450 phones for each dollar increase in the price.

Therefore, Number_of_Sales = 450 - 4 ⋅ x

Now, if no phones are sold, you set Number_of_Sales = 0 in the formula above, and obviously sales income will equal zero.

You can calculate the amount of price increase x beyond which no phones will sell. All that's left after that is to see if 302 + x < 522.

yes this is much more sensible way of going about the business of answering the problem... more straightforward.
I was thinking though...

The problem is easy to verify in a simple manner with calculator. Simply try and see, whether the new price causes the sales to plummet to 0 amount.

so the number of sales as a funcion of price increase would be thus. Am I correct in this previous statement <--- Sometimes I get confused which stuff is the function of something. Like the velocity as the function of time. If you say it that way, then I think velocity should be the Y axis, and time should be x values and in the horizontal axis. Then the graph would show how velocity increases or decreases with respect to time moving forward.

f(x) = 450- 4x

 

[SteamKing]

yes this is much more sensible way of going about the business of answering the problem... more straightforward.
I was thinking though...

The problem is easy to verify in a simple manner with calculator. Simply try and see, whether the new price causes the sales to plummet to 0 amount.

so the number of sales as a funcion of price increase would be thus. Am I correct in this previous statement <--- Sometimes I get confused which stuff is the function of something. Like the velocity as the function of time. If you say it that way, then I think velocity should be the Y axis, and time should be x values and in the horizontal axis. Then the graph would show how velocity increases or decreases with respect to time moving forward.

f(x) = 450- 4x

For the function f(x) as described above, y = f(x) would represent the number of cell phones sold and x would represent the increase or decrease in the price of each phone from the original $302 per phone.

When there is no price increase, x = 0 and f(0) = 450 phones, which agrees with the problem statement.

When the price increases, x will be a positive number and f(x) will decrease from 450 phones; when the price decreases, x will be negative and f(x) will increase beyond 450 phones, at the rate of 4 phones up or down depending on whether the price decreases or increases, respectively.

In order to find the amount of increase in the price of the phone which results in zero sales, the x-intercept of f(x) will give you that amount.

 

[late347]

For the function f(x) as described above, y = f(x) would represent the number of cell phones sold and x would represent the increase or decrease in the price of each phone from the original $302 per phone.

When there is no price increase, x = 0 and f(0) = 450 phones, which agrees with the problem statement.

When the price increases, x will be a positive number and f(x) will decrease from 450 phones; when the price decreases, x will be negative and f(x) will increase beyond 450 phones, at the rate of 4 phones up or down depending on whether the price decreases or increases, respectively.

In order to find the amount of increase in the price of the phone which results in zero sales, the x-intercept of f(x) will give you that amount.

I was wondering about one thing...

Did i actually arrive at the correct result even though I approached the problem through the turnover function instead of the amount of cellphones function?

If my original function is set as an equation such that

0= (302+x)(450-4x)

X either 112.5
Or

X -302 (as I recall)

I suppose the positive solution would be more appropriate.

Although one might say that " I will increase the price by a negative amount"
It would be silly way of saying that in a business setting.

Probably it looks like 112.5 dollars is the maximum price increase and with that price increase there will be no products sold at all.

Of course in real life if the unit cost of something would be 0 dollars per flat-screen tv...

There would be many customers... but no sustainable business for the store itself it seems.

 

[SteamKing]

I was wondering about one thing...

Did i actually arrive at the correct result even though I approached the problem through the turnover function instead of the amount of cellphones function?

If my original function is set as an equation such that

0= (302+x)(450-4x)

X either 112.5
Or

X -302 (as I recall)

I suppose the positive solution would be more appropriate.

Although one might say that " I will increase the price by a negative amount"
It would be silly way of saying that in a business setting.

You don't see it much currently, but increasing the price of something by a negative amount is also known as a "price cut", and price cuts have been known to increase sales so that a company makes more total revenue although the unit price is less. For example, computers were once so expensive that only governments and big business could afford them. Newer technology allowed smaller and cheaper computers to be made, and eventually individuals could own a computer. Even after that occurred, prices continued to decline, and more computers were sold.

Probably it looks like 112.5 dollars is the maximum price increase and with that price increase there will be no products sold at all.

Of course in real life if the unit cost of something would be 0 dollars per flat-screen tv...

There would be many customers... but no sustainable business for the store itself it seems.

It's always hard to make money by giving away free stuff.