Question

the function d(t) defines a traveler’s 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.

• the starting distance, at 0 hours, is 300 miles.
• at 2 hours, the traveler is 725 miles from home.
• at 2.5 hours, the traveler is still moving farther from home.
• at 3 hours, the distance is constant, at 875 miles.
• the total distance from home after 6 hours is 1,062.5 miles.

Explanation:

Step1: Analyze t=0

For \( t = 0 \), use \( D(t)=300t + 125 \). Substitute \( t = 0 \): \( D(0)=300(0)+125 = 125 \) miles. So the first option is wrong.

Step2: Analyze t=2

For \( t = 2 \) (since \( 0\leq t<2.5 \)), use \( D(t)=300t + 125 \). Substitute \( t = 2 \): \( D(2)=300(2)+125=600 + 125 = 725 \) miles. So the second option is correct.

Step3: Analyze t=2.5

For \( t = 2.5 \), use \( D(t)=875 \) (constant function). So the traveler is not moving (distance is constant), so the third option is wrong.

Step4: Analyze t=3

For \( 2.5\leq t\leq3.5 \), \( D(t)=875 \) (constant). So at \( t = 3 \), distance is 875 miles (constant), so the fourth option is correct.

Step5: Analyze t=6

For \( t = 6 \) (since \( 3.5

Answer:

The correct options are:

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