A theater seats 200 people and charges $10 for a ticket. At this price, all the tickets can be sold. A survey indicates that if the ticket price is increased by a peso, the number of tickets sold will decrease by 10. What price will yield the greatest revenue?
number of increases of one unit of money (pesos ? or dollars ?) be n
price = 10+n
number sold = 200-10n
revenue
= R = (10+n)(200-10n)
= 2000 + 100n - 10n^2
completing the square
= -10(n^2 - 10n + 25 -25) + 2000
= -10(n-5)^2 + 2250
vertex is (5,2250)
price should be 15
max revenue is 2250
or , by Calculus
dR/dn = 100 - 20n = 0 for a max of R
20n = 100
n = 5 , as before
Well, it seems like this theater is in quite the predicament. Let's try to solve this mirthful mathematical riddle, shall we?
To maximize revenue, we need to find the perfect price point where the number of tickets sold multiplied by the price will give us the most cheddar.
Let's break it down step by step:
1. At the current price of $10, all 200 seats are filled. So, the current revenue is $10 multiplied by 200, which equals $2000.
2. According to the survey, if the ticket price goes up by a peso (that's $1, if you're not familiar with the lingo), the number of tickets sold will decrease by 10. So, we have to factor that in.
3. Let's suppose we increase the ticket price by $1. The new price would be $11. With this increase, we can expect to sell 10 fewer tickets, which means 200 - 10 = 190 tickets will be sold.
4. Now, let's calculate the revenue at this hypothetical price point. It would be $11 multiplied by 190, which equals $2090.
5. We can keep going and run the numbers with different price increments, always taking into account that for every $1 increase, the number of tickets sold decreases by 10. To make things comical, let's do a quick summary:
Price: $10 - Tickets Sold: 200 - Revenue: $2000
Price: $11 - Tickets Sold: 190 - Revenue: $2090
Price: $12 - Tickets Sold: 180 - Revenue: $2160
And so on...
6. We keep going until we find the price point that generates the greatest revenue. The revenue starts to decrease as the number of tickets sold decreases due to the price increase.
So, my dear friend, the funniest answer to your question would be that the price yielding the greatest revenue would be the one where the theater manager stops laughing because they reached the peak revenue point. You'll have to perform the calculations and keep an eye out for that magical price point.
Remember, laughter is the best currency!
To determine the price that will yield the greatest revenue, we need to find the ticket price that maximizes the total revenue.
Let's start by analyzing the relationship between ticket price and the number of tickets sold:
- At $10 per ticket, all 200 tickets can be sold, resulting in a revenue of $10 * 200 = $2000.
- If the ticket price is increased by $1 (from $10 to $11), the number of tickets sold will decrease by 10. This means that instead of selling 200 tickets, only 200 - 10 = 190 tickets will be sold at $11 per ticket, resulting in a revenue of $11 * 190 = $2090.
To maximize the revenue, we need to continue this analysis and determine the ticket price at which the revenue is maximized.
- If the ticket price is further increased by $1 (from $11 to $12), the number of tickets sold will decrease by another 10. This means that only 190 - 10 = 180 tickets will be sold at $12 per ticket, resulting in a revenue of $12 * 180 = $2160.
By continuing this analysis, we can determine the revenue for different ticket prices:
- At $10: Revenue = $10 * 200 = $2000
- At $11: Revenue = $11 * 190 = $2090
- At $12: Revenue = $12 * 180 = $2160
- At $13: Revenue = $13 * 170 = $2210
- At $14: Revenue = $14 * 160 = $2240
- At $15: Revenue = $15 * 150 = $2250
- At $16: Revenue = $16 * 140 = $2240
- At $17: Revenue = $17 * 130 = $2210
- At $18: Revenue = $18 * 120 = $2160
From this analysis, we can observe that the revenue initially increases as the ticket price is increased, but after reaching a certain point, the revenue starts to decline.
In this case, the ticket price that yields the greatest revenue is $15, resulting in a revenue of $2250.
To determine the price that will yield the greatest revenue, we need to analyze the relationship between the ticket price and the number of tickets sold.
Let's break down the problem step by step.
1. Start with the given information:
- Theater capacity: 200 people
- Initial ticket price: $10
2. Calculate the initial revenue at the given price:
- Revenue = Ticket price x Number of tickets sold
- Number of tickets sold = Theater capacity = 200
- Revenue = $10 x 200 = $2000
3. Now, let's consider the effect of increasing the ticket price by $1:
- New ticket price: $10 + $1 = $11
- According to the survey, the number of tickets sold will decrease by 10:
- New number of tickets sold = Theater capacity - 10 = 200 - 10 = 190
- New revenue = $11 x 190 = $2090
4. Continuing the analysis, let's repeat the same process for increasing the ticket price by $2, $3, and so on.
| Price Increase | Ticket Price | Tickets Sold | Revenue |
|---------------|--------------------|--------------|-------------------------------|
| 0 | $10 | 200 | $2000 |
| 1 | $10 + $1 = $11 | 190 | $2090 |
| 2 | $10 + $2 = $12 | 180 | $2160 |
| 3 | $10 + $3 = $13 | 170 | $2210 |
| ... | ... | ... | ... |
5. From the table, we can see that increasing the ticket price gradually decreases the number of tickets sold while increasing the revenue. However, there will be a point where the revenue starts to decrease. We need to find that point to determine the price that yields the greatest revenue.
6. Calculate the revenue for each price increase until the revenue starts to decrease:
- Keep increasing the ticket price by $1 and calculating the new revenue until a decrease in revenue is observed.
| Price Increase | Ticket Price | Tickets Sold | Revenue |
|---------------|--------------------|--------------|-------------------------------|
| 0 | $10 | 200 | $2000 |
| 1 | $11 | 190 | $2090 |
| 2 | $12 | 180 | $2160 |
| 3 | $13 | 170 | $2210 |
| 4 | $14 | 160 | $2240 |
| 5 | $15 | 150 | $2250 |
| 6 | $16 | 140 | $2240 |
| 7 | $17 | 130 | $2210 |
| 8 | $18 | 120 | $2160 |
7. From the table, we see that after reaching a ticket price of $15 and selling 150 tickets, the revenue starts to decrease. This means that increasing the ticket price further would not yield the greatest revenue.
Thus, the price that will yield the greatest revenue is $15 per ticket.