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A stadium can hold 25 000 people - Leaving Cert Mathematics - Question 5 - 2013
Question 5
A stadium can hold 25 000 people. People attending a regular event at the stadium must purchase a ticket in advance. When the ticket price is €20, the expected atten... show full transcript
Worked Solution & Example Answer:A stadium can hold 25 000 people - Leaving Cert Mathematics - Question 5 - 2013
Step 1
If the ticket price was €18, how many people would be expected to attend?
Answer
When the ticket price is reduced from €20 to €18, the reduction is €2. Given that every €1 decrease results in an increase of 1000 attendees, we can calculate:
[ 12000 + (20 - 18) \times 1000 = 12000 + 2000 = 14000 ]
Thus, the expected attendance would be 14,000 people.
Step 2
Step 3
Step 4
Find the price at which tickets should be sold to give the maximum expected income.
Answer
To find the maximum expected income, we first differentiate the function f(x):
[ f'(x) = 32000 - 2000x ]
Setting this equal to zero gives:
[ 32000 - 2000x = 0 \Rightarrow x = 16 ]
Thus, the optimal ticket price is €16.
Step 5
Step 6
Suppose that tickets are instead priced at a value that is expected to give a full attendance at the stadium. Find the difference between the income from the sale of tickets at this price and the maximum income calculated at (e) above.
Answer
For full attendance at 25,000 people:
[ I_{full} = 25,000 \text{ tickets} \times 20 = 500,000 ]
With the maximum income found to be €256,000, the difference is:
[ Difference = 500,000 - 256,000 = 244,000 ]
Step 7
The stadium was full for a recent special event. Two types of tickets were sold, a single ticket for €16 and a family ticket (2 adults and 2 children) for a certain amount. The income from this event was €365,000. If 1000 more family tickets had been sold, the income from the event would have been reduced by €14,000. How many family tickets were sold?
Answer
Let the number of family tickets sold be ( y ). Then, the total number of tickets sold is:
[ 25000 - y \text{ (single tickets)} ]
The income can therefore be represented as:
[ 16(25000 - y) + p \times y = 365000 ]
Where ( p ) is the price of the family ticket. From the problem statement, if 1000 more tickets are sold:
[ 16(25000 - (y + 1000)) + p (y + 1000) = 365000 - 14000 ]
We can set up the equations and rearrange to find ( y ) with further calculations.