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Tatiana [17]
3 years ago
6

The average ticket price for a concert at the opera house was $50. The average attendance was 4000. When the ticket price was ra

ised to $52, attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize the revenue for the opera house?
Mathematics
1 answer:
kotegsom [21]3 years ago
6 0

Answer

given,

opera house ticket = $50

attendance = 4000 persons

now,

opera house ticket = $52

attendance = 3800 person

assuming these are the points on the demand curve

(x, p) = (4000,50) and (x,p) = (3800,52)

using point slope formula

p-50 = \dfrac{50-52}{4000-3800}(x - 4000)

p-50 = \dfrac{-2}{200}(x - 4000)

p-50 = \dfrac{-x}{100}+ 40

p = \dfrac{-x}{100}+ 90

R(x) = x . p

R(x) = x (\dfrac{-x}{100}+ 90)

R(x) = \dfrac{-x^2}{100}+ 90x)

\dfrac{d}{dx}(R(x)) = \dfrac{d}{dx}(\dfrac{-x^2}{100}+ 90x))

\dfrac{d}{dx}(R(x)) = (\dfrac{-2x}{100})+90)

at \dfrac{d}{dx}(R(x)) = 0

\dfrac{-2x}{100}= -90

x = 4500

\dfrac{d^2}{d^2x}(R(x)) = -ve

hence at x =4500 the revenue is maximum

for maximum revenue ticket price will be

p = \dfrac{-4500}{100}+ 90

p = $45

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The perimeter of a triangular field is 120m. Two of the sides are 21m and 40m.Calculate the largest angle of the field.
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Step-by-step explanation:

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                         B

                     ↓            ↓  

                  ↓                          ↓

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            A →  →  → →  →  → →  → →    →    →  C

Consider ABC as the triangular field;

Angle A is calculated by applying cosine rule;

a^2 = b^2 + c^2 - 2bc \ Cos A\\\\Cos \ A = \frac{b^2 + c^2 - a^2}{2bc} \\\\Cos \ A = \frac{40^2 + 59^2 - 21^2}{2 \times 40 \times 59} \\\\Cos \ A = 0.983\\\\A = Cos ^{-1} (0.983)\\\\A = 10.6 \ ^0

Angle B is calculated as follows;

Cos \ B = \frac{a^2 + c^2 - b^2}{2ac} \\\\Cos \ B = \frac{21^2 + 59^2 - 40^2}{2 \times 21 \times 59} \\\\Cos \ B = 0.937\\\\B= Cos ^{-1} (0.937)\\\\B = 20.5 \ ^0

Angle C is calculated as follows;

Cos \ C = \frac{a^2 + b^2 - c^2}{2ab} \\\\Cos \ C = \frac{21^2 + 40^2 - 59^2}{2 \times 21 \times 40} \\\\Cos \ C = -0.857\\\\C = Cos ^{-1} (-0.857)\\\\C = 149\ ^0

Therefore, the largest angle of the field is 149⁰.

       

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