At the moment OHaganBooks 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 books for $20 each, but the price is dropping at a rate of $1 per week. I need to know at what rate OHaganBooks' revenue is rising or falling given these conditions. I would also like to see the company's revenue increase at a rate of $5000 per week. At what rate would sales have to have been increasing to accomplish this?
Can somebody help me answer this and explain the steps to solve the problem?
This is a related rates problem, and all problems of this type have three distinct steps:
(1) Write the general equation for the problem. In this case, we want the equation for revenue.
(2) Implicitly differentiate the equation with respect to time. It's important to note that you should NOT plug in any known values for your variables until AFTER differentiating.
(3) Plug in what you know to solve for what you need.
Try that, and let us know if you get stuck. Good luck.
What do you mean in step 2?
For example, if you were given a triangle, your general equation would be the Pythagorean theorem... a^2 + b^2 = c^2.
Find the derivate (by implicit differentiation) for the second step. You'll be using the power and chain rules.
2a (da/dt) + 2b (db/dt) = 2c (dc/dt)
The da/dt is the "rate of change of a with respect to time."
In your problem, your equation will be revenue. Revenue = (# books)(cost of each book). I let b be the number of books and c be the cost of each book.
R = b*c
You have to use the power rule here (first multiplied by the derivate of the second plus second multiplied by the derivate of the first).
dR/dt = (b)(dc/dt) + (c)(db/dt)
Plug in what you know now by analyzing the given information. For example, let's look at this sentence: "Also, it is now selling all its books for $20 each, but the price is dropping at a rate of $1 per week." This gives us the cost of each book (c = 20) and the rate of change of the cost of each book (dc/dt = -1). Do that with all the given information, and solve for the rate of change of revenue (dR/dt).
Try that, and I'll be glad to answer any questions you have.
Sorry, the PRODUCT rule is "first * derivate of the second + second * derivate of the first." (I referred to that as the power rule.)
Okay so I got
q=1000+200x
p=20-1x
x=weeks
R=20-1x(1000+200x)
R'=3000-400x
Is this right? If so how do I find how much the revenue is rising or falling and at what rate would sales have to be increasing to accomplish $5000 revenue increase per week?
No, you're supposed to use the equation...
dR/dt = (b)(dc/dt) + (c)(db/dt)
which is the same as
R' = b(c') + c(b')
We got that from the steps above. Now plug in the information that you know to solve for dR/dt (or R') for the first part.
After you get that, we can work on the second part.
b = the number of books
c = the cost of each book
db/dt = the rate of change of the number of books sold
dc/dt = the rate of change of the cost of each book
We're given all this information in the problem, so all we have to do is plug that into our equation.
I am so confused, this is not making any sense
At the moment OHaganBooks is selling 1000 books per week that is book rate b and sales are rising at a rate of 200 books per week.that is db/dt Also, it is now selling all its books for $20 each,Pbut the price is dropping at a rate of $1 per weekdP/dt . I need to know at what rate OHaganBooks' revenue is rising or falling given these conditionsdR/dt.
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Start off with the revenue function.
Revenue= Priceperbook*books/week
dRevenue/dt= Priceperbook* dbrate/week + books/week*dP/dt
or dR/dt= P db/dt + b * dP/dt
and I have highlighted this in the words above. For info, this is not a related rates problem, but just a simple product differentiation.