You have the right idea for the first section.
The starting distance is D = 900 to represent how far Jamil is from his destination. That distance shrinks by 75 miles each hour.
That's how we arrive at D(t) = 900 - 75t
This is the same as D(t) = -75t + 900
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The first D(t) value in your table is incorrect.
If we plug in t = 0, then,
D(t) = -75t + 900
D(0) = -75*0 + 900
D(0) = 0 + 900
D(0) = 900
This tells us the starting distance is 900 miles. This is not the distance Jamil has traveled so far. It's the distance he needs to travel to get where he wants to go.
Let's follow similar steps for t = 1
D(t) = -75t + 900
D(1) = -75*1 + 900
D(1) = -75 + 900
D(1) = 825
Jamil has 825 miles left to go when reaching the 1 hour mark.
It looks like you wrote the correct value here.
If we plugged in t = 2, then we get:
D(t) = -75t + 900
D(2) = -75*2 + 900
D(2) = -150 + 900
D(2) = 750
He now has 750 miles left to go at the 2 hour marker.
You are correct here as well. Nice work.
This process is repeated until we get this table
As t goes up, D(t) goes down.
The first row represents the y-intercept.
So the location (0,900) is when Jamil first boards the train.
The graph is a straight line because both t and D(t) variables are continuous. We can have something like t = 1.75 or t = 1.78965; the decimal values can go on forever based on whatever accuracy we need.
In other words, pick any two distinct points. We can find the midpoint of them and have this midpoint make sense. This is why we have continuous variables.
Side note: Use Desmos or GeoGebra to graph. The equation to type in would be y = -75x+900
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Jamil reaches his destination when the distance D is 0 miles. Again, this is the distance from his current location to where he wants to go.
We'll replace D(t) with 0 and solve for t.
D(t) = -75t + 900
0 = -75t + 900
75t = 900
t = 900/75
t = 12
It takes exactly 12 hours for him to reach his destination.
The domain therefore is 0 ≤ t ≤ 12
t = 0 is the smallest input allowed, while t = 12 is the largest input.
Any value of t between these endpoints is valid in the domain.
We can condense 0 ≤ t ≤ 12 into the interval notation [0,12]
The square brackets include each endpoint.
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Now onto the last portion.
We plug in t = 4 to find out how far he is from his destination.
We could refer to the table, but let's write out the steps anyway.
D(t) = -75t + 900
D(4) = -75*4 + 900
D(4) = -300 + 900
D(4) = 600
Jamil is 600 miles away from his target after he has been on the train for 4 hours.
This means t = 4 leads to D(t) = 600. Refer to the table above.
We can also write it as x = 4 pairs with y = 600.