$8 per hour mowing lawns and & $12 per hour babysitting. she wants at LEAST $100 per week but can't won't more than 12 hours a weeks write a graph & system of lines inequalities

Question

8$ an hour mowing lawns and 12$ an hour babysitting. wants to make at least 100$ but can't work more than 12 hrs a week. after write and graph a system of linear inequalities finally list two possible combinations of hours she could work at each job

Answer 1

$8 per hour mowing lawns and & $12 per hour babysitting. she wants at LEAST $100 per week but can't won't more than 12 hours a weeks write a graph & system of lines inequalities

Mowing a lawn = $8/hr

Babysitting= $12/hr

Work Time = WT <= 12 hrs/wk

Income = M >= $100 / week

Let x be the number of hrs mowing lawns.

  “   y  “    “         “         “    “   babysitting.

Setting up our inequalities,

M = 6x + 12 y   à 8x + 12y >= 100,  i.e. 2x + 3y >= 25

WT = x + y     --> x + y <= 12

The system of inequalities is,

2x + 3y >= 25

x + y <= 12

x >= 0, y >= 0

These inequalities can be graphed using the lines,

l1: 2x + 3y = 25

l2: x + y = 12

and are shown below,

The inequality x + y <= 12 is given by all that area below the line l1, such that both x and y are positive, i.e. the triangle AOB.

The inequality 2x + 3y >= 25 is given by all that area above the line l2, such that both x and y are positive, i.e. the triangle COD.

The common area satisfying both requirements is the triangle ACE. Only points within the triangle ACE are both below l1 and above l2.

Income, M, is maximised for that point within triangle ACE that is furthest from the origin, which is the point A, i.e. where x = 0 and y = 12.

M = 6x + 12y 6*9 + 12*12 = 144

Max income: $144

Answer 2

Let B be the number of hours baby-sitting and M the number of hours mowing the lawn.

M+B≤12 (no more than 12 hours a week working).

8M+12B≥100 (combined earnings must be at least $100).

This reduces to 2M+3B≥25.

M is the horizontal axis and B the vertical axis.

The red line and shaded area represents M+B≤12 while the blue line and shaded area represents 2M+3B≥25.

The darker shading is where both inequalities are satisfied. All points within this dark region fulfil both inequalities.

Where the lines cross, M=11 and B=1, that is, 11 hours mowing and 1 hour baby-sitting. Earnings=88+12=$100.

Another solution: 8 hours mowing, 3 hours baby-sitting. Total time 11 hours. Earnings=64+36=$100.

Another solution: 8 hours mowing, 4 hours baby-sitting. Total time 12 hours. Earnings=64+48=$112.

Another solution: 4 hours mowing, 8 hours baby-sitting. Total time 12 hours. Earnings=32+96=$128.

Maximum earnings when M=0, 12 hours baby-sitting, earnings=$144.

Minimum time=8⅓ hours, earnings from baby-sitting and no mowing=$100.