Since the price has been increasing by $7 per month, the rate at which revenue is changing when the price is $70 is $24500/month
To answer the question, we need to know what revenue is
What is revenue?
This is the total price for the total amount of goods sold.
Revenue, R = px where
- p = price and
- x = quantity = 77000 - 5p²
So, R = px
= p(77000 - 5p²)
= 77000p - 5p³
The rate at which the revenue is increasing
To find the rate at which the revenue is increasing with respect to price, we differentiate R with respect to p.
So, dR/dp = d(77000p - 5p³)/dp
= 77000dp/dp - 5dp³/dp
= 77000 - 15p²
The rate at which the revenue is changing with the price is $70
Substituing p = 70 into dR/dp, we have
dR/dp = 77000 - 15p²
dR/dp = 77000 - 15(70)²
= 77000 - 15(4900)
= 77000 - 73500
= 3500
The rate at which the revenue is changing per month
The rate at which the revenue is changing per month is
dR/dt = dR/dp × dp/dt where dp/dt = rate at which price is changing = $7 per month.
Now, dR/dt = dR/dp × dp/dt
So, dR/dt = 3500 × $ 7/month
dR/dt = $24500/month
So, the rate at which revenue is changing when the price is $70 is $24500/month
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