Question

the data in the table represents a companys profit based on the number of items produced. which equation best represents the data?

profit based on number of items produced

items produced (x) | dollars of profit (y)
100 | -70,500
200 | 50
300 | 50,100
400 | 80,300
500 | 90,400
600 | 78,000

y = -1.026x² + 1016.402x - 162075
y = -1.036x² + 1024.771x - 163710
y = 298.214x - 66317.667
y = 196.2x - 18710

Explanation:

Step1: Substitute \(x = 100\) into each equation

• For \(y=-1.026x^{2}+1016.402x - 162075\):

\(y=-1.026\times100^{2}+1016.402\times100 - 162075=-10260 + 101640.2-162075=-70494.8\)

• For \(y=-1.036x^{2}+1024.771x - 163710\):

\(y=-1.036\times100^{2}+1024.771\times100 - 163710=-10360+102477.1 - 163710=-71592.9\)

• For \(y = 298.214x-66317.667\):

\(y=298.214\times100-66317.667=29821.4 - 66317.667=-36496.267\)

• For \(y = 196.2x-18710\):

\(y=196.2\times100-18710=19620 - 18710 = 910\)

Step2: Substitute \(x = 200\) into each equation

• For \(y=-1.026x^{2}+1016.402x - 162075\):

\(y=-1.026\times200^{2}+1016.402\times200 - 162075=- 41040+203280.4-162075=-175.6\approx50\) (approximate due to rounding in the equation coefficients)

• For \(y=-1.036x^{2}+1024.771x - 163710\):

\(y=-1.036\times200^{2}+1024.771\times200 - 163710=-41440+204954.2-163710=-2195.8\)

• For \(y = 298.214x-66317.667\):

\(y=298.214\times200-66317.667=59642.8-66317.667=-6674.867\)

• For \(y = 196.2x-18710\):

\(y=196.2\times200-18710=39240 - 18710=20530\)

Step3: Substitute \(x = 300\) into each equation

• For \(y=-1.026x^{2}+1016.402x - 162075\):

\(y=-1.026\times300^{2}+1016.402\times300 - 162075=-92340+304920.6-162075 = 50505.6\approx50100\) (approximate due to rounding in the equation coefficients)

• For \(y=-1.036x^{2}+1024.771x - 163710\):

\(y=-1.036\times300^{2}+1024.771\times300 - 163710=-93240+307431.3-163710 = 50481.3\)

• For \(y = 298.214x-66317.667\):

\(y=298.214\times300-66317.667=89464.2-66317.667 = 23146.533\)

• For \(y = 196.2x-18710\):

\(y=196.2\times300-18710=58860 - 18710=40150\)

Answer:

\(y=-1.026x^{2}+1016.402x - 162075\)