new need help,

[dloldtime]

500 students would buy a book if it were sold for $50. Also 100 more students would buy the book for each $5 reduction in the price of the book. Write an equation that expresses the total sales revenue (y), which is the cost per book times the number of books sold, as a function of the cost of each book (x).

If I layout the info
# students cost total
500 $50 $25000
600 $45 $27000
700 $40 $28000
800 $35 $27000
900 $30 $27000
1000 $25 $25000
1100 $20 $22000
1200 $15 $18000
1300 $10 $13000
1400 $5 $7000
1500 $0 $0

From this I surmise that the first part of the equation should be y = 1500x - ????. Not sure how to figure out the rest of the equation?

 

[mmm4444bot]

They assigned x = book's price? They want you to write the revenue in terms of the book's price?

I would have expected them to ask for it in terms of the number of price reductions, instead, because it's the effect of these reductions that drives both the price and the number sold.

In other words, let x = the number of $5 price reductions

Then, expressions for the book's price and the number sold are respectively:

50 - 5*x

500 + 100*x

Does this make sense?

The price goes down $5 for each reduction, and the number sold goes up 100 for each reduction.

 

[dloldtime]

You did not say what you are studying, where you are studying, or what your math background is. Those things help us know HOW to answer you in a way that is best suited to you.

Do you remember about slopes and intercepts? The standard form of a linear equation is \(\displaystyle y = mx + b,\) where m is the slope and b is the y-intercept?

For each change in x of -5, there is a change in y of +100. So \(\displaystyle m = \dfrac{100}{-5} = -20.\)

And you figured out the y-intercept, which is 1500.

So the equation is \(\displaystyle y = -20x + 1500.\)

Let's check. \(\displaystyle If\ x = 0,\ y = -20(0) + 1500 = 0 + 1500 = 1500.\) OK

\(\displaystyle If\ x = 20,\ y = -20(20) + 1500 = -400 + 1500 = 1100.\) OK

\(\displaystyle If\ x = 50,\ y = -(20)(50)+1500 = -1000 + 1500 = 500.\) OK

Jeff thanks for the reply
It makes since what you did to find the formula for the line Y = 1500 - 20x, I am still struggling because in this equation y = the number of students and I am looking for the total revenue. I now know the final answer is
y (revenue)= 1500x-20x² where x is the cost of each book. I can plug the numbers in and make it work but I am still struggling how to get to the formula. By the way, this is 11th grade math.

 

[Mrspi]

500 students would buy a book if it were sold for $50. Also 100 more students would buy the book for each $5 reduction in the price of the book. Write an equation that expresses the total sales revenue (y), which is the cost per book times the number of books sold, as a function of the cost of each book (x).

If I layout the info
# students cost total
500 $50 $25000
600 $45 $27000
700 $40 $28000
800 $35 $27000
900 $30 $27000
1000 $25 $25000
1100 $20 $22000
1200 $15 $18000
1300 $10 $13000
1400 $5 $7000
1500 $0 $0

From this I surmise that the first part of the equation should be y = 1500x - ????. Not sure how to figure out the rest of the equation?

I'd let x = number of $5 reductions in price
For each of these reductions, the number of books sold increases by 100, so 100x is the increase in sales for x price reductions of $5 each

Original price was $50
New price, after "x" reductions of $5 will be 50 - 5x

Original sales were 500 books
Since each of the x price reductions results in an additional 100 books sold, the sales AFTER x price reductions will be 500 + 100x

Revenue is price * number of sales

Revenue after x price reductions will be (50 - 5x)*(500 + 100x)

If we let y = sales revenue, then we have

y = (50 - 5x)(500 + 100x)

That's going to be a parabola...the parabola will open downward. If you are looking for maximum sales revenue, it will occur at the vertex of the parabola.

This is NOT going to be a "linear function" so it does not fit the y = mx + b pattern.

 

[dloldtime]

I was all wrong, Mrspi is correct that the equation is not linear. Now she redefined x. Leaving x the way you originally defined it, we still get a parabola. First, y does NOT equal the number of students who buy books. NOR does y equal the number of books sold. As Mrspi said revenue = price per book sold * number of books sold. In some fit of stupidity, I gave you an equation for the number of books sold = the number of students who purchased a book. Let x = the price of one book. Let y = revenue. Let z = the number of books sold. So y = zx. I calculated the number of books sold in my first post, not the revenue. z = 20x - 1500. See my first post. So y = zx = x(20x - 1500) = 20x^2 - 1500x, which is the answer in your book. Sorry for getting this wrong.

That's it. I understand it now. Thanks for the help.