To determine the best estimate for the price of a house in the year 2021 using the given quadratic equation [tex]\( p = 1.95 t^2 + 12.25 t + 125 \)[/tex], follow these steps:
1. Identify the value of [tex]\( t \)[/tex]: Since [tex]\( t \)[/tex] represents the number of years since 2010, calculate [tex]\( t \)[/tex] for the year 2021.
[tex]\[
t = 2021 - 2010 = 11
\][/tex]
2. Plug [tex]\( t = 11 \)[/tex] into the equation: Substitute [tex]\( t \)[/tex] into the quadratic equation to find [tex]\( p \)[/tex].
[tex]\[
p = 1.95(11)^2 + 12.25(11) + 125
\][/tex]
3. Calculate the square and product terms:
[tex]\[
11^2 = 121
\][/tex]
[tex]\[
1.95 \times 121 = 235.95
\][/tex]
[tex]\[
12.25 \times 11 = 134.75
\][/tex]
4. Sum the results to find [tex]\( p \)[/tex]:
[tex]\[
p = 235.95 + 134.75 + 125 = 495.7 \text{ (in thousands of dollars)}
\][/tex]
5. Convert [tex]\( p \)[/tex] to dollars: Since [tex]\( p \)[/tex] is given in thousands of dollars, convert it to dollars:
[tex]\[
p = 495.7 \times 1000 = 495,700 \text{ dollars}
\][/tex]
6. Round to the nearest thousand: To match the answers provided, round the result to the nearest thousand:
[tex]\[
495,700 \approx 496,000
\][/tex]
Therefore, the best estimate for the price of the house in year 2021 is [tex]\( \boxed{496,000} \)[/tex] dollars.
