Question

the data in the table represents a companys profit based on the number of items produced.
profit based on number of items produced

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

which equation best represents the data?

• $y = -1.026x^2 + 1016.402x - 162075$
• $y = -1.036x^2 + 1024.771x - 163710$
• $y = 298.214x - 66317.667$
• $y = 196.2x - 18710$

Explanation:

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

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

\(y=-1.026\times100^{2}+1016.402\times100 - 162075\)
\(y=-10260+101640.2 - 162075=-70494.8\approx - 70500\)

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

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

• 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 the equations (focus on the first two quadratic equations as the linear ones already deviated at \(x = 100\))

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

\(y=-1.026\times200^{2}+1016.402\times200 - 162075\)
\(y=- 41040+203280.4-162075 = 1785.4\)

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

\(y=-1.036\times200^{2}+1024.771\times200 - 163710\)
\(y=-41440+204954.2-163710 = 50.2\approx50\)

Answer:

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