A manufacturer determines that m employees will produce a total of q units each week. The relation between the number of employees and the quantity is given as: 𝑞 = 5𝑚2 + 2𝑚 + 4. Given that the demand function for the product is given as 𝑝 = 1000 𝑞+8 , determine a. The rate of change of price with respect to quantity b. The rate of change of price with respect to the number of employees c. Determine the marginal revenue when the number of employees is 15

a. The rate of change of price with respect to quantity

The rate of change of price with respect to quantity is the derivative of the price function with respect to quantity. The price function is given as p = 1000/(q+8).

To find the derivative, we can use the quotient rule, which states that the derivative of a function in the form of f(x) = g(x)/h(x) is f'(x) = (g'(x)h(x) - g(x)h'(x))/[h(x)]^2.

Applying this rule to our function, we get:

dp/dq = -1000 / (q+8)^2

b. The rate of change of price with respect to the number of employees

To find the rate of change of price with respect to the number of employees, we need to use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

The price function can be written as a composite function p(m) = 1000/(5m^2 + 2m + 12), where the outer function is f(x) = 1000/x and the inner function is g(m) = 5m^2 + 2m + 12.

The derivative of the outer function is f'(x) = -1000/x^2 and the derivative of the inner function is g'(m) = 10m + 2.

Using the chain rule, we get:

dp/dm = f'(g(m)) * g'(m) = -1000/(5m^2 + 2m + 12)^2 * (10m + 2)

c. Determine the marginal revenue when the number of employees is 15

The marginal revenue is the rate of change of total revenue with respect to the number of units sold, which is the derivative of the revenue function with respect to quantity.

The revenue function is R(q) = p*q = 1000q/(q+8).

Taking the derivative of this function with respect to quantity, we get:

dR/dq = 1000/(q+8) - 1000q/(q+8)^2

Substituting q = 515^2 + 215 + 4 into this equation, we can find the marginal revenue when the number of employees is 15.