Question

after t hours, lizas distance from home, in miles, is given by d(t)=170 + 30(t - 3). (a) what is the practical interpretation of the constants 3 and 170? (b) rewrite the function in slope - intercept form and give a practical interpretation of the constants. d(t)=i + i t the constant term is i the coefficient of t is i etextbook and media hint

Explanation:

Step1: Interpret 3 and 170 in original function

At \(t = 3\), new travel phase starts, 170 miles from home at \(t = 3\).

Step2: Expand the function

\[D(t)=170+30t - 90\]

Step3: Simplify the function

\[D(t)=80 + 30t\]

Step4: Interpret new constants

80 miles from home at \(t = 0\), 30 miles per hour speed.

Answer:

(a) The constant 3 means that after 3 hours, a new - phase of her journey starts (perhaps a change in speed or mode of travel). The constant 170 means that at \(t = 3\) hours, Liza is 170 miles from home.
(b) First, expand the function \(D(t)=170 + 30(t - 3)\):
\[

$$\begin{align*} D(t)&=170+30t-90\\ D(t)&=80 + 30t \end{align*}$$

\]
The function in slope - intercept form is \(D(t)=80+30t\). The constant term 80 means that when \(t = 0\) (the start of the time - measurement), Liza is 80 miles from home. The coefficient of \(t\) (30) means that Liza's distance from home increases by 30 miles per hour.