A local theater company is presenting a musical for kids at the community center. Each adult ticket costs $8, and each child ticket costs $4. On the opening night, the theater company needs to earn at least $1,280 from ticket sales to cover its costs. The communi
center has a capacity of 240 people.
If the solution region represents the number of adult and child tickets that needs to be sold for the theater company to cover its cost
determine which graph represents the solution set to the system of inequalities representing this situation.

Option (1)

Step-by-step explanation:

Let the number of adult tickets sold = x

And the number of child tickets sold = y

Cost of adult tickets sold = $8x

And the cost of child tickets sold = $4y

If the theater company needs to earn at least $1280,

8x + 4y ≥ 1280

2x + y ≥ 320 ---------(1)

Since community center has a capacity of 240 people,

x + y ≤ 240 ----------(2)

By graphing these inequalities with the help of a graphing tool,

Shaded area above the solid line will be represented by inequality (1) (red line)

And the shaded area below the line will be the represented by inequality (2) (blue line)

Common area of both the shaded regions will represent the solution area .

Option (1) will be the answer.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

two data sets with the same median will not have the same varaibility

step-by-step explanation: