Suppose that the revenue in dollars from the sale of x campers is given by the formula shown below.
R(x) = 58,000x + 38,000(10 + x)−1 − 4000
(a) Find the marginal revenue when 10 units are sold.
R'=58000-38000/(10+x)^2
Go figure when x=10
I'm sorry. Looking back at the question I posted, I forgot to made -1 an exponent. It should read:
R(x)= 58,000x + 38,000(10+x)^-1 - 4000
To find the marginal revenue when 10 units are sold, we need to find the derivative of the revenue function R(x) with respect to x, and then substitute x = 10 into the resulting expression.
Step 1: Derive the revenue function R(x)
To do this, we need to know the basic derivative rules. For the revenue function R(x) = 58,000x + 38,000(10 + x)^(-1) - 4000, we will apply the power rule and the constant multiple rule.
The power rule states that if f(x) = x^n, then the derivative of f(x) with respect to x is f'(x) = nx^(n-1). In our case, n = -1.
The constant multiple rule states that if f(x) = c*g(x), where c is a constant and g(x) is a differentiable function, then the derivative of f(x) with respect to x is f'(x) = c*g'(x).
Using these rules, we can find the derivative of R(x):
R'(x) = 58,000 - 38,000*(-1)(10 + x)^(-2)
Simplifying this expression further, we get:
R'(x) = 58,000 + 38,000*(10 + x)^(-2)
Step 2: Substitute x = 10 into R'(x)
To find the marginal revenue when 10 units are sold, we substitute x = 10 into the derived expression R'(x):
R'(10) = 58,000 + 38,000*(10 + 10)^(-2)
R'(10) = 58,000 + 38,000*(20)^(-2)
R'(10) = 58,000 + 38,000*(1/400)
R'(10) = 58,000 + 38,000/400
R'(10) = 58,000 + 95
R'(10) = 58,095
Therefore, the marginal revenue when 10 units are sold is $58,095.