The Revenue from the sale of x units of a product is represented by the following formula: R = 20(3x + 1)^-1 + 50x - 17 dollars. Find the Marginal Revenue when 20 units are sold?
To find the marginal revenue when 20 units are sold, we need to find the derivative of the revenue function with respect to x and evaluate it at x = 20.
Step 1: Find the derivative of the revenue function.
R = 20(3x + 1)^-1 + 50x - 17
To take the derivative, we can use the power rule and the constant rule.
dR/dx = 20(d/dx)(3x + 1)^-1 + 50
Using the power rule, we have:
dR/dx = 20(-1)(3x + 1)^-2(3) + 50
Simplifying further:
dR/dx = -60(3x + 1)^-2 + 50
Step 2: Evaluate the derivative at x = 20.
Substitute x = 20 into the derivative:
dR/dx = -60(3(20) + 1)^-2 + 50
Simplifying:
dR/dx = -60(61)^-2 + 50
dR/dx = -60/61^2 + 50
Step 3: Calculate the result.
Using a calculator, we can evaluate the expression:
dR/dx ≈ -0.160
Therefore, the marginal revenue when 20 units are sold is approximately -0.160 dollars.
To find the marginal revenue when 20 units are sold, we need to take the derivative of the revenue function with respect to the number of units sold, x, and then plug in x = 20.
The revenue function is given by R = 20(3x + 1)^-1 + 50x - 17.
First, let's find the derivative of the revenue function with respect to x:
dR/dx = 20 * d/dx [(3x + 1)^-1] + 50.
To differentiate (3x + 1)^-1, we can use the chain rule. The derivative of u^-1, where u = 3x + 1, is given by -1 * u^-2 * du/dx.
The derivative of u = 3x + 1 with respect to x is simply 3.
Therefore, the derivative of (3x + 1)^-1 with respect to x is -1 * (3x + 1)^-2 * 3.
Substituting this back into the original expression for dR/dx, we get:
dR/dx = 20 * (-1 * (3x + 1)^-2 * 3) + 50.
Simplifying further, we have:
dR/dx = -60(3x + 1)^-2 + 50.
Now, we can evaluate dR/dx at x = 20:
dR/dx = -60(3*20 + 1)^-2 + 50.
Simplifying,
dR/dx = -60(61)^-2 + 50.
Calculating this expression, we get:
dR/dx ≈ -0.0659.
Therefore, the marginal revenue when 20 units are sold is approximately -0.0659 dollars.
r'=-20/(3x+1)^2 + 50
when x=20
r'=-20/61^2 + 50