Question

for 0 ≤ t ≤ 4, the average price of a home in a city is modeled by the equation p(t)=1.3t² - 12t + 120, where p(t) is in thousands of dollars, and t is in years.
a. find p(t)
p(t)= thousands of dollars per year
b. find p(2)
p(2)= thousands of dollars per year

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $p(t)=1.3t^{2}-12t + 120$, differentiating term by term:
For the first term $1.3t^{2}$, its derivative is $2\times1.3t^{2 - 1}=2.6t$. For the second term $-12t$, its derivative is $-12$. The derivative of the constant term $120$ is $0$. So $p^\prime(t)=2.6t-12$.

Step2: Evaluate $p^\prime(2)$

Substitute $t = 2$ into $p^\prime(t)$. We have $p^\prime(2)=2.6\times2-12$. First, calculate $2.6\times2 = 5.2$. Then $p^\prime(2)=5.2-12=-6.8$.

Answer:

a. $p^\prime(t)=2.6t - 12$
b. $p^\prime(2)=-6.8$