A company selling widgets has found that the number of items sold
x depends upon the price pat which they're sold, according the equation
x=80000√2p+1 .
Due to inflation and increasing health benefit costs, the company has been increasing the price by $3 per month. Find the rate at which revenue is changing when the company is selling widgets at $210 each.
dollars per month
This is far from the right answer. The correct answer is 5862.33
If you don't know what your talking about, don't respond.
To find the rate at which revenue is changing, we need to find the derivative of the revenue function with respect to time.
Given that the number of items sold, x, depends on the price, p, according to the equation x = 80000√(2p+1), we can express the revenue function as R = xp.
We are given that the price is increasing by $3 per month, so we can express the price as p(t) = 210 + 3t, where t is the number of months.
Substituting the price function into the equation for x, we have x = 80000√(2(210 + 3t) + 1).
To find the rate at which revenue is changing, we differentiate the revenue function with respect to time:
dR/dt = (dx/dt)p + x(dp/dt).
Let's find these derivatives step by step:
1. Find dx/dt:
dx/dt = d/dt [80000√(2(210 + 3t) + 1)]
To differentiate this, we will use the chain rule and the power rule:
dx/dt = 80000 * (1/2) * (2(210 + 3t) + 1)^(-1/2) * 2(210 + 3t)' = 80000 * (1/2) * (2(210 + 3t) + 1)^(-1/2) * 3
= 80000 * 3/2 * (2(210 + 3t) + 1)^(-1/2)
= 120000 * (2(210 + 3t) + 1)^(-1/2)
2. Find dp/dt:
dp/dt = d/dt (210 + 3t) = 3
3. Substitute the values back into the revenue derivative equation:
dR/dt = (dx/dt)p + x(dp/dt)
= 120000 * (2(210 + 3t) + 1)^(-1/2) * (210 + 3t) + (80000√(2(210 + 3t) + 1))(3)
Now, we can evaluate this expression when the company is selling widgets at $210 each (t = 0):
dR/dt = 120000 * (2(210 + 3(0)) + 1)^(-1/2) * (210 + 3(0)) + (80000√(2(210 + 3(0)) + 1))(3)
= 120000 * (2(210) + 1)^(-1/2) * 210 + (80000√(2(210) + 1))(3)
= 120000 * (420 + 1)^(-1/2) * 210 + (80000√(420 + 1))(3)
= 120000 * (421)^(-1/2) * 210 + (80000√421)(3)
So, the rate at which revenue is changing when the company is selling widgets at $210 each is 120000 * (421)^(-1/2) * 210 + (80000√421)(3) dollars per month.
To find the rate at which revenue is changing, we need to calculate the derivative of the revenue function with respect to time.
First, let's express the number of items sold, x, in terms of the price, p. The given equation is:
x = 80000√(2p + 1)
Now, we need to find the revenue function, R, which is the product of the price and the number of items sold:
R = px
Substituting x from the first equation into the revenue equation, we have:
R = p(80000√(2p + 1))
Now, we can find the rate of change of the revenue with respect to time (dp/dt). Since the company is increasing the price by $3 per month, we have:
dp/dt = $3/month
To calculate dR/dt (the rate at which revenue is changing), we need to use the chain rule:
dR/dt = dR/dp * dp/dt
Let's find dR/dp by differentiating the revenue function with respect to p:
dR/dp = d(p(80000√(2p + 1)))/dp
= 80000√(2p + 1) + 80000p(1/2)(2(1/2)) / √(2p + 1)
= 80000√(2p + 1) + 20000p / √(2p + 1)
Now, we can substitute the given price p = $210 into the expression to find dR/dt:
dR/dt = (80000√(2(210) + 1) + 20000(210)) / √(2(210) + 1) * dp/dt
= (80000√(420 + 1) + 20000(210)) / √(420 + 1) * 3
= (80000√421 + 20000(210)) / √421 * 3
Calculating this expression will give us the answer.
revenue r(x) = quantity * price = x * p
Assuming the usual carelessness with parentheses, I get
r = 80000p√(2p+1)
dr/dt = 80000 * (3p+1)/√(2p+1) dp/dt
so plug in your numbers to ger
dr/dt = 80000 631/√421 * 3 ≈ $7.38M/mo