Last year, 300 people attended the Ravindra Manch Drama Club’s winter play. The ticket price was Rs 70. The advisor estimates that 20 fewer people would attend for each Rs 10 increase in ticket price. (i) What ticket price would give the most income for the Drama Club?(ii) If the Drama Club raised its tickets to this price, how much income should it expect to bring in?
Answers 2
Answer:
Given,
The number of people attending the Drama Club's winter play = 300 people.
The original ticket price= Rs.70.
Increased priced = Rs.80.
The number of people attending the Drama Club's winter play after the increase= 20 people.
To Find,
Which ticket price would give the most income to the drama club?
How much income should the club expect to bring in with the new price?
Solution,
We can simply solve the problem by calculating the amount collected in both cases.
- Amount collected with Rs.70 as ticket price= 70×300 = Rs.21,000/.
- Amount collected with Rs.80 as ticket price= 80×20= Rs.1600/-.
Out of 21000 and 1600, 21000 is greater.
Hence, the ticket price of Rs.80 will give more income to the drama club.
Now, considering the number of people remain the same, 300.
Income collected by the club with the increased price= 80× 300= Rs.24,000.
Hence, the Income collected by the club with the increased price is Rs.24,000.
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Author:
ezequielrsnv
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Answer:
i) Rs. 110
ii) Rs. 24200
Step-by-step explanation:
Let the number of times the ticket price goes up be x.
Then,
The quadratic equation formed will be
(70 + 10x)(300 - 20x)
= 21000 + 3000x - 1400x - 200x^2 = -200x^2 + 1600x +21000
Let y = -200x^2 + 1600x +21000
That is, -200x^2 + 1600x + (21000 - y) = 0 -------- (1)
a = -200, b = 1600, c = 21000 - y
x must be real since it is the number of times the ticket price goes up.
So, b^2 - 4ac >= 0
(1600)^2 - 4 * -200 * (21000 - y) >= 0
2560000 + 16800000 - 800y >= 0
-800y >= -19360000
y <= (-19360000)/(-800)
y <= 24200
Therefore, y is either less than or equal to 24200
Which means, y has maximum value at 24200.
Therefore maximum income generated will be 24200.
Now, y = 24200 in (1) gives,
-200x^2 + 1600x + (21000 - 24200) = 0
-200x^2 + 1600x - 3200 =0
-200 ( x^2 - 8x + 16) = 0
x^2 - 8x + 16 = 0
(x - 4)^2 = 0
That is, x = 4
i) Ticket price that will generate the most income for the Drama club
= 70 + 10x
= 70 + 10*4
= 70 + 40 = Rs.110
ii) Income expected to be brought in = Rs. 24200
We already obtained the maximum income above but you could verify it by substituting x = 4 in (1) and thereby, obtaining the maximum value of the quadratic equation.
I hope this helps. Please do let me know if there are errors in the procedure.
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Author:
claireekl8
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