Question

An electronics store makes a profit of $72 for every television sold and $90 for every computer sold. The manager’s target is to make at least $360 a day on sales from televisions and computers. Write a linear inequality and graph the solutions. What are three possible solutions to the problem?

Answer 1

the complete question in the attached figure

 

we have 

$72 ------------------------------------ > for every television sold

$90 ------------------------------------ > for every computer sold.

let 

x--------------------- > numbers of television sold

y--------------------- > numbers of computer sold 

then 

72x+90y >= $360 ----------- >72x/72+90y/72 >= $360 /72---- >x+1.25y >= $5

y >= (5-x)/1.25--------------see the graph in attached figure 

 

case A---------- > (5,2) (3,3)  (1,4)

72x+90y --------- > 72*5+90*2=540 >= 360----- > is solution

72x+90y --------- > 72*3+90*3=486 >= 360----- > is solution

72x+90y --------- > 72*1+90*4=432 >= 360----- > is solution

 

case B------------ >  (4,0) (2,2)  (1,1)

72x+90y --------- > 72*4+90*0=288 < 360----- > is not solution

72x+90y --------- > 72*2+90*2=324 < 360----- > is not solution

72x+90y --------- > 72*1+90*1=162 < 360----- > is not solution

 

case C------------ >  (3,1) (2,2)  (1,0)

72x+90y --------- > 72*3+90*1=306 < 360----- > is not solution

72x+90y --------- > 72*2+90*2=324 < 360----- > is not solution

72x+90y --------- > 72*1+90*0=72 < 360----- > is not solution

 

case D------------ >  (4,0) (3,3)  (1,4)

72x+90y --------- > 72*4+90*0=288 < 360----- > is not solution

72x+90y --------- > 72*3+90*3=486 >= 360----- > is solution

72x+90y --------- > 72*1+90*4=432 >= 360----- > is solution

 

the answer is case A 

Answer 2

Answer: A. (5, 2), (3, 3), and (1, 4) are three possible solutions.

Explanation:

This is the set of answer choices for this problem:

A. (5, 2), (3, 3), and (1, 4) are three possible solutions.
B. (4, 0), (2, 2), and (1, 1) are three possible solutions.
C. (3, 1), (2, 2), and (1, 0) are three possible solutions. .
D. (4, 0), (3, 3), and (1, 4) are three possible solutions.

This is how you solve it.

1) name the variables:

t: number of tv sold
c: number of computer sold

2) model the profit = 72t + 90c

3) set the inequality:

The manager's target is to make at least $360 => profit ≥ 360

=> 72t + 90c ≥ 360

simplify dividing by 9 => 8t + 10c ≥ 40

simplify dividing by 2 => 4t + 5c ≥ 20

4) graph 4t + 5x ≥ 20

- draw the line 4t + 5c = 20

   notice that 5c = - 4t + 20 => c = (-4/5)t + 4

=> slope -4/5, c intercept 4

So, the solution is the region above the line 4t + 5c = 20, in the first quatrant ( t≥0, and c≥0)

5) use the set of answer choices to probe which are solutions:

A. (5, 2), (3, 3), and (1, 4) are three possible solutions.

Use the inequality to probe each pair:

- point (5,2): 4(5) + 5(2) = 20 + 10 = 30 ≥ 36 => ok

-  point (3,3): 4(3) + 5(3) = 12 + 15 = 27 ≥ 36 => ok

- point (1,4): 4(1) + 5(4) = 4 + 20 = 24 ≥ 20 => ok

Therefore that is the answer.

Check how the other choices are discarded.

B. (4, 0), (2, 2), and (1, 1) are three possible solutions.

- point (1,1) is not solution: 4(1) + 5(1) = 4 + 5 = 9 which is not ≥ 20 => not solution.

C. (3, 1), (2, 2), and (1, 0) are three possible solutions. .


- point (1,0) is not solution: 4(1) + 5(0) = 4 + 0 = 4 which is not ≥ 20 => not solution.

D. (4, 0), (3, 3), and (1, 4) are three possible solutions.

- point (4,0) is not solution: 4(4) + 5(0) = 16 + 0 = 16 which is not  ≥ 20.