Question

An accessories company finds that the revenue, in dollars, from the sale of x belts is given by R(x)=35x 10 9 ​ . Find the rate at which average revenue is changing when 176 belts have been produced and sold. First, find the rate at which the average revenue is changing when x belts have been produced. R ˉ ′ (x)=

Answer 1

The rate at which the average revenue is changing when x belts have been produced is 0.

Explanation:

To find the rate at which the average revenue is changing, we need to find the derivative of the average revenue function R(x).

The average revenue function R(x) is given by R(x) = R(x) / x, where R(x) is the revenue function.

In this case, the revenue function is R(x) = 35x * 10^9.

Substituting this into the average revenue function, we have,

R(x) = (35x * 10^9) / x.

To find the derivative of this function, we can apply the quotient rule.

The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.

Applying the quotient rule to R(x), we have:

R'(x) = ((35 * 10^9) * x - (35x * 10^9) * 1) / x^2.

Simplifying this expression, we have:

R'(x) = (35 * 10^9 - 35 * 10^9) / x^2.

R'(x) = 0 / x^2.

Therefore,

The rate at which the average revenue is changing, R'(x), is equal to 0.