Question

A theater owner wants to divide a 3600 seat theater into three​ sections, with tickets costing ​$60​, ​$90​, and ​$100​, depending on the section. He wants to have twice as many $ 60 tickets as the sum of the other​ tickets, and he wants to earn ​$256,000 from a full house. Find how many seats he should have in each section.

Answer 1

Answer:

2400 in the section of cost $60 for each ticket

800 in the section of cost $90 for each ticket

400 in the section of cost $100 for each ticket

Step-by-step explanation:-

- A theater owner wants to divide a 3600 seat theater into 3​ sections

- The tickets of the sections costing $60​, ​$90​, and ​$100

- He wants to have twice as many $ 60 tickets as the sum of the

  other​ tickets

- He wants to earn ​$256,000 from a full theater

- Assume that x represents the number of tickets cost $60, y represents

 the number of tickets cost $90, and z represents the number of tickets

 cost $100

∵ The total number of tickets = 3600

∴ x + y + z = 3600 ⇒ (1)

∵ The number of $60 tickets is twice the sum of the other tickets

∴ x = 2(y + z)

- divide both sides by 2

∴ $\frac{x}{2}=y+z$ ⇒ (2)

- Substitute equation (2) in equation (1)

∴ $x+\frac{x}{2}=3600$

∴ 1.5x = 3600

- Divide both sides by 1.5

∴ x = 2400

∵ He wants to earn ​$256,000 from a full theater

∵ x cost $60 each

∵ y cost $90 each

∵ z cost $100 each

∴ 60x + 90y + 100z = 256000

∴ 60(2400) + 90y + 100z = 256000

∴ 144000 +90y + 100z = 256000

- Subtract both sides by 144000

∴ 90y + 100z = 112000

- Divide both sides by 10 to simplify it

∴ 9y + 10z = 11200 ⇒ (3)

- Substitute value of x in equation (2)

∴ $\frac{2400}{2}=y+z$

∴ 1200 = y + z

∴ y + z = 1200 ⇒ (4)

- Solve equations (3) , (4) to find y and z

- Multiply equation (4) by -10 to eliminate z

∴ -10y - 10z = -12000 ⇒ (5)

- Add equations (3) and (5)

∴ -y = -800

- Multiply both sides by -1

∴ y = 800

- Substitute the value of y in equation (4) to find z

∴ 800 + z = 1200

- subtract both sides by 800

∴ z = 400

∴ The number of seats in each section:

   2400 in the section of cost $60 for each ticket

   800 in the section of cost $90 for each ticket

   400 in the section of cost $100 for each ticket