The function D(t) defines a traveler's distance from home, in miles, as a function of time, in hours. D(t) = StartLayout enlarge

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The function D(t) defines a traveler's distance from home, in miles, as a function of time, in hours. D(t) = StartLayout enlarge

d left-brace 1st Row 1st column 300 t + 125 , 2nd column 0 less-than-or-equal-to t less-than 2.5 2nd Row 1st column 880, 2nd column 2.5 less-than-or-equal-to t less-than-or-equal-to 3.5 3rd Row 1st column 75 t + 612.5, 2nd column 3.5 less-than t less-than-or-equal-to 6 EndLayout 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 875 miles from home. At 3 hours, the distance is constant, at 880 miles. The total distance from home after 6 hours is 1,062.5 miles

Mathematics

2 answers:

nalin [4] · Answer 1 · 2023-12-09T19:03:12+03:00

- At 2 hours, the traveler is 725 miles from home.

- At 3 hours, the distance is constant, at 880 miles. --> TRUE.

- The total distance from home after 6 hours is 1,062.5 miles --> TRUE.

Step-by-step explanation:

The function D(t) is defined as follows:

$D(t) = 300t+125$ for $t< 2.5$

$D(t) = 880$ for $2.5 \leq 3.5$

$D(t) = 75t+612.5$ for $t\leq 6$

Where

t is the time in hours

D(t) is the distance covered, in miles, after t hours

Now let's analyze the different statements:

- The starting distance, at 0 hours, is 300 miles. --> FALSE. In fact, if we substitute t = 0 into the 1st equation, we get

$D(0) = (300)(0)+125 = 125$

So, the distance at t = 0 is 125 miles.

- At 2 hours, the traveler is 725 miles from home. --> TRUE. In fact, if we substitute t = 2 into the 1st equation,

$D(2) = (300)(2)+125 = 725$

- At 2.5 hours, the traveler is 875 miles from home. --> FALSE. In fact, for t=2.5 we have to use the 2nd equation, which states that the distance is:

D(t) = 880

So, not 875 miles.

- At 3 hours, the distance is constant, at 880 miles. --> TRUE. This is clearly visible from the 2nd equation: for t between 2.5 and 3.5 (so, in this case), the distance is

D(t) = 880

- The total distance from home after 6 hours is 1,062.5 miles --> TRUE. In fact, if we replace t = 6 into the last equation,

$D(6)) = 75(6)+612.5=1062.5$