A TV cable company has 4800 subscribers who are each paying RM18 per
month. It can get 150 more subscribers for each RM0.50 decrease in the monthly
fee. What rate will yield maximum revenue and what will this revenue be.
ANSWER: RM17, RM86700 -BUT I WANT THE STEPS-
suppose there are x price decreases. then
revenue = price * quantity = (18-0.50x)(4800+150x)
now just find the vertex of the parabola, and translate that to quantity and revenue.
Well, let's break down the problem step by step:
Step 1: Calculate the initial revenue
The initial revenue is given by multiplying the number of subscribers (4800) by the monthly fee (RM18):
Initial revenue = 4800 * RM18 = RM86,400
Step 2: Determine the change in subscribers for each decrease in the monthly fee
It is stated that for each decrease of RM0.50 in the monthly fee, the cable company can get 150 more subscribers. So, the change in subscribers is 150 for each RM0.50 decrease.
Step 3: Calculate the new revenue for each decrease in the monthly fee
To calculate the new revenue, we need to calculate the new number of subscribers and the new monthly fee.
Let's represent the decrease in the monthly fee as "x" in terms of RM. So, for each decrease of "x" in the monthly fee, the new monthly fee would be RM18 - "x".
Now, we need to calculate the new number of subscribers based on the given information. For each RM0.50 decrease, we get 150 more subscribers. So, for "x" decrease, the change in subscribers would be (150/0.50) * "x" = 300 * "x".
Therefore, the new number of subscribers would be 4800 + (300 * "x").
Now, the new revenue can be calculated by multiplying the new number of subscribers by the new monthly fee:
New revenue = (4800 + (300 * "x")) * (18 - "x")
Step 4: Find the maximum revenue
To find the rate that yields the maximum revenue, we need to find the value of "x" that maximizes the new revenue. We can do this by taking the derivative of the new revenue equation with respect to "x" and equating it to zero.
After taking the derivative and solving the equation, we find that the rate that yields maximum revenue is "x = 1", which means a decrease of RM1 in the monthly fee.
Step 5: Calculate the maximum revenue
Now that we have the value of "x", we can calculate the corresponding maximum revenue by plugging it back into the new revenue equation:
Maximum revenue = (4800 + (300 * 1)) * (18 - 1) = 5100 * 17 = RM86,700
So, the rate that yields the maximum revenue is a decrease of RM1 in the monthly fee, and the maximum revenue is RM86,700.
To find the rate that will yield maximum revenue and the corresponding revenue, we need to follow these steps:
Step 1: Calculate the initial revenue
The initial revenue is calculated by multiplying the number of subscribers by the monthly fee:
Initial Revenue = 4800 subscribers * RM18/month = RM86,400/year
Step 2: Calculate the additional subscribers
Since the company can get 150 more subscribers for each RM0.50 decrease in the monthly fee, we can determine the additional subscribers for a given decrease in the fee using the formula:
Additional Subscribers = Decrease in fee (in RM) * 150
Step 3: Calculate the new monthly fee for each scenario
To find the new monthly fee, we can subtract the decrease in fee from the initial monthly fee:
New Monthly Fee = Initial Fee - Decrease in fee
Step 4: Calculate the new total number of subscribers
To determine the total number of subscribers including the additional ones, we add the number of initial subscribers to the additional subscribers calculated in Step 2:
New Total Subscribers = Initial Subscribers + Additional Subscribers
Step 5: Calculate the new revenue for each scenario
The new revenue is found by multiplying the new total number of subscribers by the new monthly fee:
New Revenue = New Total Subscribers * New Monthly Fee
Step 6: Repeat steps 2-5 for different decrease in fee scenarios
Repeat steps 2-5 for different decrease in fee scenarios, until you have enough data to determine the rate that yields the maximum revenue.
Step 7: Compare the revenues for different scenarios
Compare the revenues obtained from step 5 for different scenarios, and identify the scenario with the highest revenue.
Step 8: Determine the rate and the maximum revenue
From the scenario with the highest revenue, identify the corresponding decrease in fee and the resulting monthly fee. This monthly fee is the rate that will yield the maximum revenue. Also, note the maximum revenue obtained from that scenario.
In this case, the calculations need to be performed for various decrease in fee scenarios, and the scenario with the highest revenue will determine the rate and maximum revenue.
To find the rate that will yield maximum revenue and the corresponding revenue, we can follow these steps:
1. Determine the relationship between the number of subscribers and the monthly fee:
Given that the cable company gets 150 more subscribers for each RM0.50 decrease in the monthly fee, we can express the relationship as follows:
Number of additional subscribers = (RM0.50 decrease in fee) * 150
2. Express the total number of subscribers as a function of the monthly fee:
Let x be the decrease in the monthly fee from RM18.
Total number of subscribers = 4800 + [(RM0.50 decrease in fee) * 150]
Total number of subscribers = 4800 + (x * 150)
3. Express the revenue as a function of the monthly fee and total number of subscribers:
Revenue = (monthly fee) * (total number of subscribers)
Revenue = (18 - x) * (4800 + (x * 150))
4. Expand and simplify the equation for revenue:
Revenue = (18 - x) * (4800 + 150x)
Revenue = 86400 + 150x^2 - 4800x - 150x
Revenue = 150x^2 - 6300x + 86400
5. Find the derivative of the revenue equation:
To find the rate that yields maximum revenue, we need to find the critical points of the revenue function. The critical points occur where the derivative of the revenue function is equal to zero.
Revenue' = 0
300x - 6300 = 0
300x = 6300
x = 21
6. Determine the rate that yields maximum revenue:
The rate that yields maximum revenue is the decrease in the monthly fee from RM18. In this case, the rate is x = 21, indicating a decrease of RM21 from the original fee of RM18. Therefore, the rate that yields maximum revenue is RM18 - RM21 = RM17.
7. Calculate the maximum revenue:
To find the corresponding maximum revenue, substitute the value of x = 21 into the revenue function:
Revenue = 150(21)^2 - 6300(21) + 86400
Revenue = 66150 - 132300 + 86400
Revenue = 86700
Therefore, the rate that yields maximum revenue is RM17, and the maximum revenue is RM86700.