weirdal607
Answered

LOTS OF POINTS, PLEASE ANSWER ALL OF THEM, ESPECIALLY C.

Sarah has $8 and wants to buy a combination of cupcakes and fudge to feed at least four siblings. Each cupcake costs $2, and each piece of fudge costs $1.

This system of inequalities models the scenario:

2x + y ≤ 8
x + y ≥ 4

Part A: Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)

Part B: Is the point (8, 10) included in the solution area for the system? Justify your answer mathematically. (3 points)

Part C: Choose a point in the solution set and interpret what it means in terms of the real-world context. (3 points)



Answer :

rvkacademic

Answer:

Part A: Graph attached

Part B: No, (8,10) is  outside the shaded solution area

Part C: Point choice is (1, 5) inside the solution area. The real world context is that if Sarah buys 1 cupcake and 5 fudges, she can spend $8 or less(specifically $7) and still feed 4 siblings(specifically 6)

Step-by-step explanation:

The system of inequalities is

2x + y ≤ 8

x + y ≥ 4

We must also have x ≥ 0 and y ≥ 0 since negative values are not allowed

PART A

The inequalities can be plotted using any graphing calculator. See the attached graph. The dark shaded region bounded by points A, B, C is the feasible region for the system of inequalities

The individual points A, B and C were computed as follows

We can find the corner points of this system of equations by treating them as equalities and solving for x, y

2x + y ≤ 8 ==> 2x + y = 8

For x = 0, y = 8 so (0,8) is a corner point (point B)

For y = 0, x = 4 so (4,0) is another corner point (point A)

For x + y ≥ 4 ==> x + y = 4

Corner points are (0, 4) and (4,0) (points A and C)

Another corner point can be found by solving the equations

2x + y = 8  (1)
 x + y = 4     (2)

(1) - (2) ==> x = 4 and therefore y = 0 so again C(4,0)  features as a corner point

Note that (4,0) appears as a corner point in multiple places

Part B:

There are two ways to determine if the point (8, 10) is included in the solution space. Plug in these values of x = 8, y = 10 and see if they satisfy both equations.

For 2x + y ≤ 8 , plugging in x = 8, y = 10 gives
2(8) + 10 = 26 and this is much greater than the right side value of 8

So (8, 10) is not a solution set for the system

(The other approach is to view the point in the graph. You will see it is outside the shaded region indicating it is not a feasible solution set

Part C:

Choose a point with integer coordinates. That is obvious since a, y have to be integers - you cant buy 1/3rd of a cupcake or 3/4 of a fugde

I chose point (1, 5) which is well within the shaded feasible area

Real world context
This means that if Sarah buys 1 cupcake and 5 fudges she will keep the total cost at $8 or below and still be able to feed 4 or more siblings

Indeed,

Using x = 1, y = 5 :

2(1) + 5 = 7 which is ≤ 8

1 + 5 = 6 which is ≥ 4


The single graph I have attached to my answer contains the feasible shaded region, the infeasible point(8,10) of Part B and a feasible point(1,5) for Part C


Please ask if further clarifications are needed.

View image rvkacademic
semsee45

Answer:

A)  See below

B)  Not included

C)  (2, 3)

Step-by-step explanation:

Given system of inequalities:

[tex]\begin{cases}2x + y \leq 8\\\;\;x + y \geq 4\end{cases}[/tex]

Part A

Rewrite both equations in slope-intercept form:

  • [tex]y \leq -2x+8[/tex]
  • [tex]y \geq -x+4[/tex]

When graphing inequalities:

  • < or > : dashed line.
  • ≤ or ≥ : solid line.
  • < or ≤ : shade under the line.
  • > or ≥ : shade above the line.

The system of inequalities represents how many cupcakes and fudge Sarah can buy. As she cannot buy negative amounts of cupcakes and fudge, the graph should display the positive direction of the x axis and y axis only.

To graph the given system of inequalities:

  • Graph the line y = -2x + 8 with a solid line. Shade under the line.
  • Graph the line y = -x + 4 with a solid line.  Shade above the line.

The solution set is any point contained in the overlapping shaded region.

Part B

Point (8, 10) is not included in the solution area since it is not contained in the overlapping shaded region.

To determine if point (8, 10) is included in the solution area for the system mathematically, substitute (8, 10) into both inequalities:

[tex]\begin{aligned}\implies 2x+y &\leq4\\2(8)+10 &\leq 8\\ 16+10 & \leq8\\26&\leq8 \leftarrow \textsf{false}\end{aligned}[/tex]

[tex]\begin{aligned}\implies x+y&\geq4\\8+10 &\geq 4\\ 18 &\geq4 \leftarrow \textsf{true}\end{aligned}[/tex]

Hence proving that the point (8, 10) is not included in the solution area.

Part C

Chose a point from the overlapping shaded region of the graph where the x and y values are integers.

  • (2, 3)

Point (2, 3) means that Sarah can buy 2 cupcakes and 3 pieces of fudge.

View image semsee45
View image semsee45

Other Questions