Recall the following information from Class Ex. #2 on page 348.
"The hockey club had 7200 season ticket holders who each paid $1400 for a package of season tickets. The owner had suggested raising the price to generate more revenue, but knew that the number of season ticket holders would be reduced."
The general manager suggested that more revenue might be obtained by decreasing the price and thus attracting more fans to buy a package of season tickets. The research company, that the owner hired to explore the general manager's suggestion, reported that for every $50 decrease in price, approximately 400 new season ticket holders would be generated.
a) if the price decrease is to be a multiple of $50, determine the following:
i) the price of a package of season tickets which would generate maximum revenue
ii) the number of season ticket holders which would be generated
iii) the revenue which would be generated if the plan in a) was implemented
Can you please show your work and explain your answer? (I have the answer key but no idea how to do it, I was away on a field trip when this lesson was taught). Thank you.
"The hockey club had 7200 season ticket holders who each paid $1400 for a package of season tickets. The owner had suggested raising the price to generate more revenue, but knew that the number of season ticket holders would be reduced."
The general manager suggested that more revenue might be obtained by decreasing the price and thus attracting more fans to buy a package of season tickets. The research company, that the owner hired to explore the general manager's suggestion, reported that for every $50 decrease in price, approximately 400 new season ticket holders would be generated.
a) if the price decrease is to be a multiple of $50, determine the following:
i) the price of a package of season tickets which would generate maximum revenue
ii) the number of season ticket holders which would be generated
iii) the revenue which would be generated if the plan in a) was implemented
Can you please show your work and explain your answer? (I have the answer key but no idea how to do it, I was away on a field trip when this lesson was taught). Thank you.
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Let x be the multiple of $50 by which the price is decreased.
That is, the new price is 1400 - 50x,
and the new number of season ticket holders is 7200 + 400x.
The revenue will be
y = (7200 + 400x) (1400 - 50x)
= 2000 (18 + x) (28 - x)
= 2000 (-x^2 + 10x + 504)
= -2000 (x^2 - 10x - 504)
= -2000 (x^2 - 10x + 25) + 10,580,000 [completing the square]
= -2000 (x - 5)^2 + 10,580,000
This revenue function has a maximum of $10,580,000
with x=5, representing a $250 reduction in the price.
i) The new price is $1400 - $250 = $1150
ii) There would then be 7200 + 2000 = 9200 season ticket holders.
iii) The revenue generated is $1150 * 9200 = $10,580,000 as noted above.
Verifying the maximum:
A $300 reduction would produce revenue of $1100 * 9600 = $10,560,000.
A $200 reduction would produce revenue of $1200 * 8800 = $10,560,000.
Both are less than the revenue produced by a $250 reduction.
That is, the new price is 1400 - 50x,
and the new number of season ticket holders is 7200 + 400x.
The revenue will be
y = (7200 + 400x) (1400 - 50x)
= 2000 (18 + x) (28 - x)
= 2000 (-x^2 + 10x + 504)
= -2000 (x^2 - 10x - 504)
= -2000 (x^2 - 10x + 25) + 10,580,000 [completing the square]
= -2000 (x - 5)^2 + 10,580,000
This revenue function has a maximum of $10,580,000
with x=5, representing a $250 reduction in the price.
i) The new price is $1400 - $250 = $1150
ii) There would then be 7200 + 2000 = 9200 season ticket holders.
iii) The revenue generated is $1150 * 9200 = $10,580,000 as noted above.
Verifying the maximum:
A $300 reduction would produce revenue of $1100 * 9600 = $10,560,000.
A $200 reduction would produce revenue of $1200 * 8800 = $10,560,000.
Both are less than the revenue produced by a $250 reduction.