Marginal revenue the weekly revenue from the sale of emeralds at Eduardo’s Emerald Emporium (EEE) is decreasing at a rate of $500 per $1 increase in price and the price is decreasing at a rate of $0.45 per additional emerald sold. What is the marginal revenue?
I am not really sure what to do or how to set this up please walk me through the steps.
I am just not seeing how to set this up.
weekly revenue rate = -500*1
Price decrease rate = -.45
Marginal Revenue = (-500*1)(-.45) = 225
Well, let's start by breaking down the given information. We know that the marginal revenue, which is the change in revenue for each additional unit sold, is decreasing at a rate of $500 per $1 increase in price. This means that for each $1 increase in price, the revenue decreases by $500.
We also know that the price is decreasing at a rate of $0.45 per additional emerald sold. This means that for each additional emerald sold, the price decreases by $0.45.
To find the marginal revenue, we need to consider the relationship between price and revenue. The revenue is the product of the price and the quantity sold. So, we can express revenue as R = p * q, where R represents revenue, p represents price, and q represents quantity sold.
Now, we want to find the rate at which revenue is changing with respect to quantity sold, or the marginal revenue. To do this, we can take the derivative of the revenue function with respect to quantity sold (dq/dt) and multiply it by the quantity sold (q) to get the marginal revenue (dR/dt).
Let's start by finding the derivative of the revenue function with respect to quantity sold (dq/dt). Since the price is decreasing at a rate of $0.45 per additional emerald sold, the derivative of the price with respect to quantity sold (dp/dq) would be -0.45.
Next, we can express the derivative of the revenue function with respect to quantity sold (dq/dt) as the derivative of (p * q) with respect to quantity sold (dq/dt).
Applying the product rule, we get dq/dt = q * dp/dq + p * dq/dt. Since dp/dq is -0.45, we can substitute it into the equation.
dq/dt = q * (-0.45) + p * dq/dt.
Now, we can rearrange the equation to solve for dq/dt.
dq/dt - p * dq/dt = -0.45 * q
Factoring out dq/dt, we have:
dq/dt * (1 - p) = -0.45 * q
Finally, we can isolate dq/dt to find the marginal revenue:
dq/dt = (-0.45 * q) / (1 - p)
So, the marginal revenue is given by (-0.45 * q) / (1 - p).
I hope this explanation didn't give you too much of a headache! Remember, if you need any more help, don't hesitate to ask. I'm here to lighten the mood and assist you with your questions!
To find the marginal revenue, we need to find the rate of change of revenue with respect to quantity or price. In this case, we are given the rate of change of revenue with respect to price and the rate of change of price with respect to quantity.
Step 1: Identify the given information:
- The rate of change of revenue with respect to price is -$500 per $1 increase in price.
- The rate of change of price with respect to quantity is -$0.45 per additional emerald sold.
Step 2: Determine the relationship between revenue, price, and quantity:
We know that revenue is the product of price and quantity, so we can write the equation:
Revenue = Price x Quantity
Step 3: Differentiate the revenue equation with respect to price:
By using the product rule of differentiation, we can find the rate of change of revenue with respect to price:
dRevenue/dPrice = d(Price x Quantity)/dPrice
Step 4: Apply the differentiation rules:
Since the quantity is independent of the price, the derivative of the Revenue with respect to the Price is simply Quantity:
dRevenue/dPrice = Quantity
Step 5: Use the given information to find the marginal revenue:
Given that the rate of change of price with respect to quantity is -$0.45 per additional emerald sold, we can substitute this value in the equation we found in step 4:
dRevenue/dPrice = Quantity = -$0.45
Therefore, the marginal revenue is -$0.45 per additional emerald sold.
To find the marginal revenue in this scenario, we need to consider the relationship between price, quantity, and revenue.
Step 1: Understand the given variables and their relationships.
In this case, we are given two pieces of information:
1. Marginal revenue decreases at a rate of $500 per $1 increase in price.
2. Price decreases at a rate of $0.45 per additional emerald sold.
Step 2: Define the variables.
Let's assign the variables as follows:
- MR (Marginal Revenue): The change in revenue resulting from a change in quantity sold or price.
- P (Price): The cost of one emerald in dollars.
- Q (Quantity): The number of emeralds sold.
Step 3: Set up the equation.
Marginal revenue can be calculated using the following equation:
MR = (∂R/∂Q) * (∂Q/∂P)
Step 4: Calculate the partial derivatives.
First, let's calculate ∂R/∂Q:
Since revenue is the product of price and quantity, we can differentiate this equation with respect to quantity:
∂R/∂Q = P
Next, let's calculate ∂Q/∂P:
We know that price decreases at a rate of $0.45 per additional emerald sold. So, the change in quantity with respect to price (∂Q/∂P) is -0.45.
Step 5: Substitute the values into the equation.
MR = (∂R/∂Q) * (∂Q/∂P)
= P * (-0.45)
Step 6: Substitute the given information into the equation.
From the given information, we know that the marginal revenue decreases at a rate of $500 per $1 increase in price. So, the change in price with respect to marginal revenue is -1/500.
Hence, P = -1/500.
Substituting the values into the equation:
MR = (-1/500) * (-0.45)
Simplifying:
MR = 0.0009 or $0.0009
Therefore, the marginal revenue is $0.0009 per additional emerald sold.