A company selling widgets has found that the number of items sold x
depends upon the price p at which they're sold, according the equation
x=70000 /(√3p+1) .
Due to inflation and increasing health benefit costs, the company has been increasing the price by $4 per month. Find the rate at which revenue is changing when the company is selling widgets at $180 each.
revenue r = price * quantity = p*x = 70000p/√(3p+1)
dr/dt = 70000 * (3p+2)/(3p+1)^(3/2) dp/dt
so plug in your numbers for p and dp/dt
To find the rate at which revenue is changing, we need to take the derivative of the revenue function with respect to time. The revenue is calculated by multiplying the number of items sold by the price:
Revenue = x * p
First, let's find the derivative of the number of items sold with respect to the price (dx/dp). We can do this by using the quotient rule:
dx/dp = [d(70000)/(dp) * (√3p + 1) - 70000 * d(√3p + 1)/(dp)] / (√3p + 1)^2
Let's simplify this:
dx/dp = [0 * (√3p + 1) - 70000 * (3/2) * (√3p + 1)] / (√3p + 1)^2
= -105000√3 / (2(√3p + 1)^2)
Now, let's find the derivative of the revenue with respect to time (dt). Since the price is increasing by $4 per month, dp/dt = 4:
d(revenue)/dt = dx/dp * dp/dt
= -105000√3 / (2(√3p + 1)^2) * 4
Substituting the given price (p = $180):
d(revenue)/dt = -105000√3 / (2(√3*180 + 1)^2) * 4
= -105000√3 / (2(540√3 + 1)^2) * 4
Now, we can calculate the rate at which revenue is changing by plugging in the values into the equation.
To find the rate at which revenue is changing, we need to take the derivative of the revenue function with respect to time. However, since the revenue function depends on both the price and the number of items sold, we need to use the chain rule.
Let's first express the revenue as a function of time. Given that the price increases by $4 per month, we can express the price as a function of time t using the equation p = 180 + 4t.
Next, we can express the number of items sold as a function of the price using the equation x = 70000 / (√(3p) + 1), and substitute the price function into it:
x = 70000 / (√(3(180 + 4t)) + 1).
Finally, we can express the revenue as a function of time by multiplying the price and the number of items sold:
R(t) = x * p.
To find the rate at which revenue changes with respect to time, we need to compute dR/dt, the derivative of R(t) with respect to t:
dR/dt = d(x * p)/dt.
Using the product rule, we have:
dR/dt = dx/dt * p + x * dp/dt.
Now, let's differentiate x = 70000 / (√(3(180 + 4t)) + 1) with respect to t to find dx/dt:
dx/dt = d(70000 / (√(3(180 + 4t)) + 1))/dt.
To simplify the differentiation, let's first find d(√(3(180 + 4t))) / dt. Then, we can use the chain rule to determine dx/dt by applying the quotient rule.
d(√(3(180 + 4t)))/dt = (1/2√(3(180 + 4t))) * d(3(180 + 4t))/dt
= (1/2√(3(180 + 4t))) * 12
= 6/√(3(180 + 4t)).
Now, let's compute dx/dt using the chain rule and the quotient rule:
(dx/dt) = d(70000 / (√(3(180 + 4t)) + 1))/dt
= [d(70000)/dt * (√(3(180 + 4t)) + 1) - 70000 * d(√(3(180 + 4t)) + 1)/dt] / (√(3(180 + 4t)) + 1)^2
= [0 * (√(3(180 + 4t)) + 1) - 70000 * (6/√(3(180 + 4t))) * 4] / (√(3(180 + 4t)) + 1)^2
= -2800000/((√(3(180 + 4t)) + 1)^2 * √(3(180 + 4t))).
Now let's substitute dx/dt and the given price p = 180 into the expression for dR/dt:
dR/dt = dx/dt * p + x * dp/dt
= (-2800000/((√(3(180 + 4t)) + 1)^2 * √(3(180 + 4t)))) * 180 + (70000 / (√(3(180 + 4t)) + 1)) * 4.
Finally, we can plug in the value of the time t for which the company is selling widgets at $180 each (p = 180) and calculate the rate at which revenue is changing.