Question

Bonita is testing prototype model rocket engines. To be considered a successful launch, the rocket must reach a height of at least 24 ft and must go a horizontal distance of no more than 10 ft. The vertical height must be at least three times as high as the horizontal distance. No rocket should go higher than 33 ft.

The graph shows the feasible region, where x represents the horizontal distance and y represents the vertical distance.

Which ordered pairs meet all the constraints for a successful launch and make sense in context of the situation?

Select each correct answer.



(0, 33)

(4, 36)

(4.8, 30.5)

(9, 26)

(2, 22)

Answer 1

The ordered pairs that meet all the constraints for a successful launch and make sense in context of the situation are:

(0,33) and (4.8,30.5)

Explanation:

Here x represent the horizontal distance and y represent the vertical distance.

From the given information in the question we can make the following inequalities:

• The rocket must reach a height of at least 24 ft and must go a horizontal distance of no more than 10 ft.

y ≥ 24

and x ≤ 10

• The vertical height must be at least three times as high as the horizontal distance.

i.e. y ≥ 3x

• No rocket should go higher than 33 ft.

i.e. y ≤ 33

Hence, after combining all the inequalities we have:

x ≤ 10

24 ≤ y ≤ 33

and y ≥ 3x

1)

(0, 33)

As (0,33) satisfies all the constraints of the given problem.

Hence, this ordered pair will lie in the feasible region.

2)

(4, 36)

As the height that is the y-value has to be less than 33.

Hence, the ordered pair (4,36) does not lie in the feasible region.

3)

(4.8, 30.5)

It satisfies all the inequalities of a successful launch.

Hence, this ordered pair will lie in the feasible region.

4)

(9, 26)

As the property that y ≥ 3x mjust be satisfied.

But here at x=9

y ≥ 27

but we are given y=26.

Hence, this ordered pair also does not lie in the feasible region.

5)

(2, 22)

As the minimum vertical distance must be 24.

But here we are given y=22.

Hence, this ordered pair does not lie in the feasible region.

Answer 2

Let

x--------> the horizontal distance in feet

y--------> the vertical height in feet

we know that

The constraints are

1)
`y\geq 3x`

2)
`24\ ft\leq y \leq 33\ ft`

3)
`x\leq10 \ ft`

Let's verify each case to determine the solution to the problem.

case A) point
`(0,33)`

Substitute the values of x and y in each of the constraints

constraint 1

`33\geq 3*0`

`33\geq 0` ---------> is true

constraints 2

`24\ ft\leq 33 \leq 33\ ft`-------> is true

constraints 3

`0\leq10 \ ft`--------> is true

therefore

the ordered pair
`(0,33)` meet all the constraints for a successful launch

case B) point
`(4,36)`

Substitute the values of x and y in each of the constraints

constraint 1

`36\geq 3*4`

`36\geq 12` ---------> is true

constraints 2

`24\ ft\leq 36 \leq 33\ ft`-------> is not true

constraints 3

`4\leq10 \ ft`--------> is true

therefore

the ordered pair
`(4,36)` does not meet all restrictions for a successful launch

case C) point
`(4.8,30.5)`

Substitute the values of x and y in each of the constraints

constraint 1

`30.5\geq 3*4.8`

`30.5\geq 14.4` ---------> is true

constraints 2

`24\ ft\leq 30.5 \leq 33\ ft`-------> is true

constraints 3

`4.8\leq10 \ ft`--------> is true

therefore

the ordered pair
`(4.8,30.5)` meet all the constraints for a successful launch

case D) point
`(9,26)`

Substitute the values of x and y in each of the constraints

constraint 1

`26\geq 3*9`

`26\geq 27` ---------> is not true

constraints 2

`24\ ft\leq 26 \leq 33\ ft`-------> is true

constraints 3

`9\leq10 \ ft`--------> is true

therefore

the ordered pair
`(9,26)` does not meet all restrictions for a successful launch

case E) point
`(2,22)`

Substitute the values of x and y in each of the constraints

constraint 1

`22 \geq 3*2`

`22\geq 6` ---------> is true

constraints 2

`24\ ft\leq 22 \leq 33\ ft`-------> is not true

constraints 3

`2\leq10 \ ft`--------> is true

therefore

the ordered pair
`(2,22)` does not meet all restrictions for a successful launch

the answer is

`(0,33)`

`(4.8,30.5)`