A club currently has
300
members who pay
$ 400
per month for membership dues. The club's board members want to increase monthly revenue by lowering the monthly dues in hopes of attracting new members. A market research study has shown that for each
$ 1 decrease in the monthly membership price, an additional
3 people will join the club. What price should the club charge to maximize revenue? What is the maximum revenue?
Suppose there are x price decreases.
so, price is 400-x
membership is 300+3x
revenue is thus (400-x)(300+3x)
= -3x^2+900x+120000
So, just find the vertex of the parabola to determine price and max revenue.
Nsj
To find the price that will maximize revenue, we need to consider the relationship between the number of members and the monthly dues. Let's break down the problem step-by-step:
Step 1: Calculate the current monthly revenue:
Current monthly revenue = Number of members * Monthly dues
Current monthly revenue = 300 * $400
Current monthly revenue = $120,000
Step 2: Determine the relationship between price and the number of new members:
For each $1 decrease in the monthly membership price, an additional 3 people will join the club. So, the number of new members is:
Number of new members = $1 decrease in price * 3
Step 3: Calculate the number of members at different price levels:
Number of members at a lower price = Current number of members + Number of new members
Number of members at a lower price = 300 + ($1 decrease in price * 3)
Step 4: Calculate the monthly revenue at different price levels:
Monthly revenue at a lower price = Number of members at a lower price * Price
Monthly revenue at a lower price = (300 + ($1 decrease in price * 3)) * Price
Step 5: Find the price that maximizes revenue:
To find the price that maximizes revenue, we need to differentiate the monthly revenue equation with respect to price and set it equal to zero, and solve for price.
d(Monthly revenue at a lower price) / d(Price) = 0
Let's calculate the derivative:
d(Monthly revenue at a lower price) / d(Price) = 3 * (300 + ($1 decrease in price * 3)) - ($1 decrease in price * 3)
0 = 3 * (300 + ($1 decrease in price * 3)) - ($1 decrease in price * 3)
Simplifying:
0 = 900 + ($3 decrease in price) - ($3 decrease in price)
0 = 900
Since the derivative is constant and never equal to zero, we don't have a maximum or minimum point. This means the revenue keeps increasing as the price decreases.
However, we have to consider practical constraints. The price cannot be decreased indefinitely as it may reach a level where it becomes unprofitable.
So, to attract new members while also maximizing revenue, the club needs to lower the price to a point where members are incentivized to join without causing a significant loss in revenue.
To find the optimal price, we need more information about the market demand and cost structure of the club.
To determine the price that the club should charge to maximize revenue, we need to find the point where the marginal revenue equals zero. Marginal revenue is the added revenue gained from each additional member. In this case, for every $1 decrease in membership price, 3 additional members join the club.
Let's break down the problem step by step to find the optimal price point:
1. Start with the given information:
- Current number of members = 300
- Current membership price = $400
- Additional members per $1 decrease = 3
2. Calculate the new number of members for different price decreases:
- For a $1 decrease: additional members = 3, total members = 300 + 3 = 303
- For a $2 decrease: additional members = 2 * 3 = 6, total members = 300 + 6 = 306
- For a $3 decrease: additional members = 3 * 3 = 9, total members = 300 + 9 = 309
3. Calculate the total revenue for each scenario:
- Revenue at $400: $400 * 300 = $120,000
- Revenue at $399: $399 * 303 = $121,197
- Revenue at $398: $398 * 306 = $121,788
- Revenue at $397: $397 * 309 = $122,673
4. Determine the change in revenue for each price decrease:
- Change in revenue from $400 to $399: $121,197 - $120,000 = $1,197
- Change in revenue from $399 to $398: $121,788 - $121,197 = $591
- Change in revenue from $398 to $397: $122,673 - $121,788 = $885
5. Calculate the marginal revenue for each price decrease:
- Marginal revenue from $400 to $399: $1,197 / 1 = $1,197
- Marginal revenue from $399 to $398: $591 / 1 = $591
- Marginal revenue from $398 to $397: $885 / 1 = $885
6. Analyze the marginal revenue results:
- Since the marginal revenue decreases as the price decreases, we need to continue reducing the price until marginal revenue reaches zero or starts to decrease.
7. Continue analyzing with additional price decreases until marginal revenue starts to decrease:
- For a $4 decrease: additional members = 4 * 3 = 12, total members = 300 + 12 = 312, revenue: $312 * ($400 - $4) = $118,848
- For a $5 decrease: additional members = 5 * 3 = 15, total members = 300 + 15 = 315, revenue: $315 * ($400 - $5) = $124,200
8. Determine the optimal price point and maximum revenue:
- Based on the analysis, the maximum revenue occurs at a $5 price decrease, so the club should charge $395.
- The maximum revenue would then be $124,200.
So, to maximize revenue, the club should charge $395 per month, resulting in a maximum revenue of $124,200.