Question

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the function d(t) defines a travelers distance from home, in miles, as a function of time, in hours.
d(t) =
\

$$\begin{cases} 300t + 125, & 0 \\leq t < 2.5 \\\\ 875, & 2.5 \\leq t \\leq 3.5 \\\\ 75t + 612.5, & 3.5 < t \\leq 6 \\end{cases}$$

which times and distances are represented by the function? select three options.
\square the starting distance, at 0 hours, is 300 miles.
\square at 2 hours, the traveler is 725 miles from home.
\square at 2.5 hours, the traveler is still moving farther from home.
\square at 3 hours, the distance is constant, at 875 miles.
\square the total distance from home after 6 hours is 1,062.5 miles.
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Explanation:

Step1: Check starting distance (t=0)

Use \(D(t)=300t + 125\). \(D(0)=300(0)+125=125\) (not 300).

Step2: Check t=2 hours

Use \(D(t)=300t + 125\). \(D(2)=300(2)+125=725\) (correct).

Step3: Check t=2.5 hours

Interval \(2.5 \leq t \leq 3.5\): \(D(t)=875\) (constant, not moving).

Step4: Check t=3 hours

Interval \(2.5 \leq t \leq 3.5\): \(D(t)=875\) (constant, correct).

Step5: Check t=6 hours

Use \(D(t)=75t + 612.5\). \(D(6)=75(6)+612.5=1062.5\) (correct).

Answer:

B. At 2 hours, the traveler is 725 miles from home., D. At 3 hours, the distance is constant, at 875 miles., E. The total distance from home after 6 hours is 1,062.5 miles.