the gym charges 180$ for a yearly membership. there are currently 1000 members. for every 5$ increase the gym will lose 10 members. how much should the gym charge to maximize its revenue?
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To maximize its revenue, the gym needs to find the price that will result in the highest number of members while still generating significant revenue. One way to approach this is to analyze how the change in price affects the number of members and total revenue.
Let's start by calculating the initial revenue with a yearly membership fee of $180 and 1000 members:
Initial revenue = ($180 * 1000) = $180,000
Now, we need to determine the relationship between the membership fee increase and the corresponding decrease in the number of members.
Given:
For every $5 increase in membership fee, the gym loses 10 members.
This means that for every $5 decrease in membership fee, the gym gains 10 members.
Now, we can determine the number of members lost as the price increases:
Number of members lost = (Price increase in $) / ($5) * 10
To maximize revenue, we can assume that the number of members lost should be equal to the number of members gained as the price decreases:
Number of members lost = Number of members gained
(Price increase in $) / ($5) * 10 = Number of members gained
Now, let's represent the new number of members as (1000 - X) and the price increase as Y:
(Y / $5) * 10 = (1000 - X)
Now we have an equation that relates the price increase to the number of members lost. Rearranging this equation gives us:
X = 1000 - (Y / $5) * 10
Now we can determine the revenue at each price point by multiplying the number of members by the membership fee:
Revenue = (1000 - X) * (180 + Y)
Substituting X from the equation above:
Revenue = (1000 - (Y / $5) * 10) * (180 + Y)
To find the price that maximizes revenue, we can determine the value of Y that results in the maximum revenue. We can achieve this by finding the derivative of the revenue function with respect to Y and setting it to zero, and then solving for Y.
Differentiating the revenue function with respect to Y gives us:
dRevenue/dY = 0
Solving this equation will give us the value of Y that maximizes revenue.
To determine the price the gym should charge to maximize its revenue, we need to find the price that will result in the highest number of members while also considering the decrease in membership for each price increase.
Let's break down the problem step by step:
1. Calculate the current revenue: Since there are 1000 members and the annual membership fee is $180, the gym's current revenue is 1000 * $180 = $180,000.
2. Determine the relationship between price and membership: The problem states that for every $5 increase in price, the gym will lose 10 members. This implies that for every $5 decrease in price, the gym will gain 10 members.
3. Calculate the membership change for a price change: Since the current price is $180 and there will be a $5 increase for every 10-member loss, we can divide the price change by $5 to find the number of 10-member losses. So, $5 / $5 = 1. This means that for every $5 increase, there will be a loss of 10 members.
4. Calculate the membership for different prices: Let's calculate the number of members for different prices to determine the price that maximizes revenue.
a. For the current price of $180, there are 1000 members.
b. For a $5 increase in price to $185, there will be a loss of 10 members, resulting in 1000 - 10 = 990 members.
c. For a $10 increase in price to $190, there will be a loss of 20 members, resulting in 1000 - 20 = 980 members.
Continuing this pattern, we can calculate the number of members for different prices.
5. Determine the revenue at each price: To determine the revenue at each price, we multiply the price by the number of members.
a. At the current price of $180, the revenue is $180 * 1000 = $180,000.
b. At a price of $185, the revenue is $185 * 990 = $183,150.
c. At a price of $190, the revenue is $190 * 980 = $186,200.
Calculating the revenue for other prices will give us a complete picture.
6. Analyze the revenue: After calculating the revenue for different prices, we can see that the revenue increases with the price up to a certain point and then starts to decrease. To find the price that maximizes revenue, we need to identify the peak value of the revenue.
From the calculations above, we can see that the revenue increases as the price increases up to $190, at which point it starts to decrease if the price continues to increase.
Therefore, to maximize revenue, the gym should charge $190 for its annual membership.
current charge -- $180
membership ---- 1000
Let the number of $5 increases be m
new charge = 180 + 5n
membership = 1000 - 10n
revenue = (180+5n)(1000-10n)
= 180000 + 3200 - 50n^2
= -50(n^2 - 64n - 3600)
this is a parabola and the vertex will yield our answer
the n of the vertex is 64/2 = 32
so the new charge should be 180 + 5(32) = 340
for a membership of 1000 - 10(32) = 680
check:
current revenue = 180(1000) = 180,000
at "best", revenue = 340(680) = 231,000
my answer is reasonable