Assume that a demand equation is given by q=9000-100p. Find the marginal revenue for the given production levels.

a. 500 Units
the marginal revenue at 500 units is

revenue is demand*price

r = p(9000-100p) = 9000p - 100p^2
marginal revenue is
dr/dp = 9000-200p

To find the marginal revenue at a given production level, you need to take the derivative of the demand equation with respect to quantity (q) and multiply it by the production level. The derivative of the demand equation q = 9000 - 100p with respect to q is simply -100.

Now, substitute the given production level of 500 units into the derivative to calculate the marginal revenue.

MR = (-100) * 500
MR = -50,000

Therefore, the marginal revenue at a production level of 500 units is -50,000.

To find the marginal revenue at 500 units, we need to calculate the derivative of the demand equation with respect to price (p).

The given demand equation is q = 9000 - 100p.

To find the marginal revenue, we need to find the derivative of the demand equation, in other words, the rate at which revenue is changing with respect to the quantity sold.

Let's start by differentiating the demand equation with respect to p:

dq/dp = -100

Now, it's important to note that marginal revenue (MR) is the change in total revenue (TR) divided by the change in quantity (q). In other words:

MR = ∆TR / ∆q

In this case, since we are looking for the marginal revenue at 500 units, we need to determine the change in revenue (∆TR) when the quantity changes from 500 units (q) to 501 units (q+∆q), assuming the price remains constant.

Let's substitute q = 500 into the demand equation:

q = 9000 - 100p
500 = 9000 - 100p
100p = 9000 - 500
100p = 8500
p = 8500 / 100
p = 85

So, at q = 500 units, the price is p = 85.

Next, we need to determine the change in revenue when the quantity changes from 500 units to 501 units (q+∆q). Let's substitute q = 501 into the demand equation:

q = 9000 - 100p
501 = 9000 - 100p
100p = 9000 - 501
100p = 8499
p = 8499 / 100
p = 84.99

So, at q = 501 units, the price is p = 84.99.

Now, we can calculate the change in revenue (∆TR) by subtracting the total revenue at 500 units from the total revenue at 501 units:

∆TR = (q+∆q) × p - q × p
∆TR = 501 × 84.99 - 500 × 85

Finally, we can calculate the marginal revenue (MR) by dividing ∆TR by ∆q (which is 1 in this case):

MR = ∆TR / ∆q
MR = (∆TR) / (1)

Calculating the value, we get:

MR = (501 × 84.99 - 500 × 85) / 1
MR ≈ -84.99

Therefore, the marginal revenue when producing 500 units is approximately -84.99.