Question

1topThe table below shows the yearly average selling price of a home in the Oxford Woods subdivision during certainyears. Using a quartic model, approximate in the selling price of a home in Orchard Woods in 2020.YearAverageSelling Price1998 $144,0002003 $174,6002010 $175,1202012 $180,5002015 $203,000estionSelect one:O a. $300,303O b. $306,128O c. $310,502O d. $315,457

Answer 1

We have a general quartic model for the average selling price (S) as a function of the number of years passed since 1998 (x):

S(x) = a*x⁴ + b*x³ + c*x² + d*x + e

We can calculate the value of x for every year:

1998: x = 0

2003: x = 5

2010: x = 12

2012: x = 14

2015: x = 17

2020: x = 22

Now, we have the equations (given by the table):

S(0) = 144 000

S(5) = 174 600

S(12) = 175 120

S(14) = 180 500

S(17) = 203 000

Or:

For x = 0:

a*0⁴ + b*0³ + c*0² + d*0 + e = 144 000 => e = 144 000 ...(1)

For x = 5:

a*5⁴ + b*5³ + c*5² + d*5 + 144 000 = 174 600

625a + 125b + 25c + 5d = 30 600

Dividing by 5:

125a + 25b + 5c + d = 6120 ...(2)

For x = 12:

a*12⁴ + b*12³ + c*12² + d*12 + 144 000 = 175 120

20736a + 1728b + 144c + 12d = 31 120

Dividing by 4:

5184a + 432b + 36c + 3d = 7780 ...(3)

For x = 14:

a*14⁴ + b*14³ + c*14² + d*14 + 144 000 = 180 500

19208a + 1372b + 98c + 7d = 18250 ...(4)

For x = 17:

a*17⁴ + b*17³ + c*17² + d*17 + 144 000 = 203 000

17⁴a + 17³b + 17²c + 17d = 59000 ...(5)

Solving (2), (3), (4), and (5) for a, b, c, and d:

a = -(151/3213)

b = 187006/3213

c = -(4763263/3213)

d = 12941200/1071

Now, for the year 2020, we use these numbers and x = 22:

S(22) = -(151/3213)*22^4 + (187006/3213)*22^3 - (4763263/3213)*22^2 + (12941200/1071)*22 + 144 000

S(22) = $301 039.259