Question

A band is playing at an auditorium with floor seats and balcony seats. The band wants to sell the floor tickets for $15 each and balcony tickets for $12 each. They want to make at least $3,000 in ticket sales.

Below is a sketch of the graph.
a) Write the inequality of the line.

b) Identify the ordered pair for point A.

c) Identify the ordered pair for point B.

d) State which direction they should shade and why that makes sense in context.

Answer 1

Explanation:

Lets denote number of floor tickets by x and

Lets denote number of balcony tickets by y as shown in the figure in the question

so, floor tickets are sold for $15 each which means number of x tickets would be sold by $15 multiplied by x which means 15x and balcony tickets are sold for $12 each which means number of y tickets would be sold by $12 multiplied by y which means 12y furthermore it says they want to make at least $3000 which would mean they want to make at least $3000 or higher meaning ≥3000 so the inequality would be

a)
`15x+12y\geq 3000`

b) The ordered pair A we can see that A is the y-intercept which means the point A can also be written as (0 , y) so we put this point into our equation

15x + 12y = 3000 which is the boundary line. So here goes,

`15x+12y=3000\\15(0)+12(y)=3000\\12y=3000\\y=3000/12\\y=250\\`

so our y-intercept is 250 which means Point A is (0 , 250)

c) They should shade the area above the boundary line because in this context the band that is playing wants to make at least $3000 which means they want to make more than $3000 . So when a ≥ sign appear we shade the area above the boundary line and when the ≤ sign appears we shade the area below the boundary line. If it were in the question the bands wants to make at most $3000 then the ≤ would appear.

I have attached a graph to clearly visualize the solution as well.