A company's revenue from selling x units of an item is given as R=2000x−3x^2. If sales are increasing at the rate of 50 units per day, how rapidly is revenue increasing (in dollars per day) when 110 units have been sold?
dR/dt = (2000 - 6x) dx/dt
now plug in your numbers
To find the rate at which revenue is increasing when 110 units have been sold, we need to differentiate the revenue function with respect to time and then substitute the given values.
The revenue function is given as R = 2000x - 3x^2, where x represents the number of units sold.
Differentiating this function with respect to time (t), we get:
dR/dt = d/dt (2000x - 3x^2)
To find the rate at which sales are increasing per day, we can differentiate x with respect to t:
dx/dt = 50 units/day (given)
Now, let's substitute x = 110 units and dx/dt = 50 units/day into the derivative of the revenue function:
dR/dt = d/dt (2000x - 3x^2)
dR/dt = 2000(dx/dt) - 6x(dx/dt)
dR/dt = 2000(50) - 6(110)(50)
Simplifying the expression:
dR/dt = 100000 - 33000
dR/dt = 67000
Therefore, the revenue is increasing at a rate of $67,000 per day when 110 units have been sold.
To find the rate at which revenue is increasing, we need to take the derivative of the revenue function with respect to time.
First, let's rewrite the revenue function: R(x) = 2000x - 3x^2.
Now we can find the derivative: dR/dt = d(2000x - 3x^2)/dt.
Note that the rate at which sales (x) is changing with time (t) is given as 50 units per day.
So, dx/dt = 50 units/day.
Taking the derivative of the revenue function with respect to time, we get: dR/dt = d(2000x - 3x^2)/dt = 2000*dx/dt - 6x*dx/dt.
Substituting dx/dt = 50 into the equation, we get: dR/dt = 2000*(50) - 6x*(50).
Now we can find the value of x when 110 units have been sold and substitute it into the equation: dR/dt = 2000*(50) - 6*(110)*(50).
Calculating the expression, we find: dR/dt = 100,000 - 330,000 = -230,000 dollars per day.
Therefore, when 110 units have been sold, the revenue is decreasing at a rate of $230,000 per day.
given :
dx/dt = 50
R = 2000x - 3x^2
dR/dt = 2000 dx/dt - 6x dx/dt , so when x = 110
dR/dt = 2000(110) - 6(110)(50)
= .....
make sure to include the units in your answer, I skipped them for simpler typing