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=10000√(6p+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 $100 each.
___ dollars per month
x=10000√(6p+1)
That makes no sense.
As the price goes up you sell more?
Is it a typo?
maybe
x=10000 / √(6p+1)
Yes, you did it right, sorry it didn't transfer properly over.
x=10000 / √(6p+1)
dx/dt = -30000/(6p+1)^(3/2) dp/dt
To find the rate at which revenue is changing, we need to first express the revenue as a function of the number of items sold and the price at which they are sold. Revenue is calculated by multiplying the quantity sold (x) by the price per item.
We are given the equation x = 10000√(6p+1), which represents the number of items sold as a function of the price. To find revenue, we multiply this function by the price per item (p):
Revenue = x * p
Substituting the given equation for x, we have:
Revenue = (10000√(6p+1)) * p
We want to find the rate at which revenue is changing when the company is selling widgets at $100 each. So, we substitute p = 100 into the revenue equation:
Revenue = (10000√(6(100)+1)) * 100
Revenue = 10000√(601) * 100
Now, to find the rate at which revenue is changing, we need to differentiate the revenue function with respect to time (since we are interested in the rate of change over time). In this case, the independent variable is the price (p), and the rate at which the price is changing is $4 per month. So, we take the derivative of the revenue function with respect to p and then multiply it by the rate of change of the price (dp/dt = 4):
d(Revenue)/dt = d(Revenue)/dp * dp/dt
To find d(Revenue)/dp, we differentiate the revenue function with respect to p using the chain rule:
d(Revenue)/dp = 10000 * √(601) + 10000 * 0.5 * (601)^(-0.5) * 6
Simplifying this expression, we find:
d(Revenue)/dp = 10000 * √(601) + 300000/√(601)
Now, we can plug in the values and calculate the rate at which revenue is changing:
Rate of revenue change = (10000 * √(601) + 300000/√(601)) * 4
Simplifying further, we get:
Rate of revenue change ≈ (40000√(601) + 1200000/√(601)) dollars per month