# At what rate is revenue rising or falling.

• Nov 29th 2009, 12:46 PM
colinn
At what rate is revenue rising or falling.
Problem:

"For the past few months, OHaganBooks.com has seen its weekly sales increase. At the moment OHaganBooks.com is selling 1000 books per week and its sales are rising at a rate of 200 books per week. Also, it is now selling all its book for \$20 each, but the price is dropping at a rate of \$1 per week. I need to know at what rate OHaganBooks.com's revenue is rising or falling given these conditions. I would also like to see that company's revenue increase at a rate of \$5,000 per week. At what rate would sales have to have been increasing to accomplish this?"

So those are the two questions I have to answer. Pretty simple I'm guessing. I understand (R=pq) which is not hard to calculate for each week. I put the answers out in a table showing the difference in revenue from the previous week, and the total revenue for the week.

However, I'm guessing there is some calculus technique I need to use here and I'm not quite sure what it is. If anyone could take a look at it and see what they think I would greatly appreciate it.
• Nov 29th 2009, 01:38 PM
tom@ballooncalculus
The rates to relate are of R, p and q, and since R=pq we can see that differentiating them all with respect to t will involve the product rule. I would think...

http://www.ballooncalculus.org/asy/diffProd/rates2.png

... where

http://www.ballooncalculus.org/asy/prod.png

... is the product rule - straight continuous lines differentiating downwards (integrating up) with respect to t.

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• Nov 29th 2009, 01:55 PM
Stev381
Here is what I think,

Let P(x) = the profit, they are selling there books 20\$ a piece.
Let f(x) = the number of books they are selling per week.

f(x)=1000+200x where x is the week number.

But they sell books 20\$ but with price dropping of 1\$ per week per book.

P(x)=1000(20)-1000*x+200(20)*x-200*x

I need to know at what rate OHaganBooks.com's revenue is rising or falling given these conditions:

Take the derivative of P(x)

P'(x)=-1000+4000-200=2800\$

Hence revenue are rising at a rate of 2800\$ per week.

At what rate would sales have to have been increasing to accomplish this?"

You want the derivative of our function = 5 000\$ per week.
1000 books per week sold is fix...let u denote the rising sales.
Then under the same condition:

P(x)=1000*20-1000*x+u*20*x-u*x
P'(x)=-1000+20*u-u=5 000

Solve for u you get: 315.79 books, hence sales have to go up of
316 books per week.

• Nov 29th 2009, 04:55 PM
colinn
I appreciate the above answers, but I'm still not understanding the equations here I guess.
• Nov 29th 2009, 07:46 PM
Stev381
Ok,

well this company is selling 1000 books a week. Moreover, the sales are raising by 200 books per week. That is, f(x)=number of book sold in one week.

f(x)=1000+200x where x is the week number.

They are selling the books 20\$ each and the price of them is deacreasing by 1\$ dollar a week. That is,

f(x)=1000*(20\$)-1000*x+200*x*(20)-200*x where x is the week number.

In this problem, the price dropping is = to the week number...