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?
write the functions.
revenue= number books sold * price book
dRev/dt=n dp/dt + p dn/dt
you are given n, p and dp/dt, and dn/dt
calculate.
For the second part, set dRev/dt to 5000
find dn/dt given n, p, dp/dt
I will be happy to critique your work.
ok i got this
R=(1000+200x)(20-x)
R'=3000-400x
is that right so far?
and x=weeks?
To determine the rate at which OHaganBooks' revenue is rising or falling, we need to consider the factors affecting it:
1. Sales: At the moment, OHaganBooks is selling 1000 books per week, and this is increasing at a rate of 200 books per week.
2. Price: Currently, each book is being sold for $20, and this price is dropping at a rate of $1 per week.
To calculate OHaganBooks' revenue, we multiply the number of books sold by the price per book. Let's denote the number of books sold as S, the price per book as P, and the revenue as R.
R = S * P
Given the information, we can proceed to find the rate of change of revenue.
1. Rate of change of sales (dS/dt):
We are given that sales are increasing at a rate of 200 books per week. Therefore, dS/dt = 200.
2. Rate of change of price (dP/dt):
We know that the price is dropping at a rate of $1 per week. Therefore, dP/dt = -1.
3. Rate of change of revenue (dR/dt):
To find this, we use the product rule of differentiation:
dR/dt = S * dP/dt + P * dS/dt
Substituting the given values:
dR/dt = 1000 * (-1) + 20 * 200
Simplifying the expression:
dR/dt = -1000 + 4000
dR/dt = 3000
Therefore, OHaganBooks' revenue is rising at a rate of $3000 per week.
Next, let's determine the rate at which sales should increase to achieve a revenue increase of $5000 per week. Let's denote the desired revenue increase as ΔR and the required rate of change of sales as ΔS/Δt.
We can rearrange the previous equation to solve for ΔS/Δt:
ΔS/Δt = (ΔR - P * dS/dt) / dP/dt
Given ΔR = $5000, P = $20, dS/dt = 200, and dP/dt = -1:
ΔS/Δt = ($5000 - $20 * 200) / (-1)
ΔS/Δt = ($5000 - $4000) / (-1)
ΔS/Δt = -$1000 / (-1)
ΔS/Δt = 1000
Therefore, sales would have to increase at a rate of 1000 books per week to achieve a revenue increase of $5000 per week.
I hope this explanation helps you understand the steps to solve the problem.