Question

(a) what is the distance traveled at 0 hours?
□ miles
(b) what is the distance traveled at 1 hour?
□ miles
(c) how much does the distance increase for each hour since frank crossed the bridge?
□ miles
(d) are the amounts given in parts (b) and (c) equal? why or why not? choose the best answer.
○ yes, because the line passes through (0, 0).
○ yes, because the line does not pass through (0, 0).
○ no, because the line passes through (0, 0).
○ no, because the line does not pass through (0, 0).

Explanation:

Step1: Analyze part (a)

At 0 hours (the start time), if we assume a linear relationship (since it's about distance over time with a constant rate implied later), the distance traveled is typically 0 miles because at time \( t = 0 \), no time has passed, so distance \( d = 0 \).

Step2: Analyze part (b)

For a linear motion with constant speed, if we assume the speed is \( v \) miles per hour, at \( t = 1 \) hour, the distance \( d = v\times1 \). But since we know from part (a) that at \( t = 0 \), \( d = 0 \), and if the rate is constant, at \( t = 1 \), the distance is equal to the rate. Let's assume the rate is \( r \), so at 1 hour, distance is \( r \) miles. But since we can infer from the context (and part (d) hint) that the line passes through \( (0,0) \), the distance at 1 hour is equal to the rate (increase per hour).

Step3: Analyze part (c)

The distance increase per hour is the slope of the distance - time graph. For a line passing through \( (0,0) \) and \( (1, r) \), the slope (rate of increase) is \( \frac{r - 0}{1 - 0}=r \) miles per hour.

Step4: Analyze part (d)

If the line passes through \( (0,0) \), the equation of the line is \( d=rt \), where \( r \) is the rate. So at \( t = 1 \), \( d = r \) (part (b)), and the rate of increase (part (c)) is \( r \). So they are equal because the line passes through \( (0,0) \), so the distance at 1 hour (part (b)) is equal to the rate of increase (part (c)).

Answer:

(a) \( \boldsymbol{0} \) miles
(b) Let the rate be \( r \), if the line passes through \( (0,0) \), at 1 hour, distance is \( \boldsymbol{r} \) (but typically, if we assume a common case like speed of, say, 60 mph, but from the context, since it's a linear relationship through (0,0), the distance at 1 hour is equal to the rate. But since we can infer from part (d) that the line passes through (0,0), and for a linear function \( d = vt \), at \( t = 1 \), \( d=v \), and the increase per hour is \( v \). So if we assume a general case, the answer for (b) is equal to the answer for (c). But since we need to fill in, let's assume a simple case where the rate is, for example, if we take the standard case where at 0 hours 0 miles, at 1 hour \( r \) miles, and increase per hour \( r \) miles. But to match the options in (d), we know that when the line passes through \( (0,0) \), the distance at 1 hour (part (b)) and the increase per hour (part (c)) are equal.

(c) The same as the distance at 1 hour (from part (b)) because of the linear relationship through \( (0,0) \). If we assume the rate is \( r \), then it's \( \boldsymbol{r} \) miles (same as part (b)).

(d) The correct option is: Yes, because the line passes through \( (0, 0) \).

(Note: Since the problem is about distance - time relationships, which is a topic in Mathematics (specifically Algebra or Geometry - coordinate geometry). The above steps are based on the properties of linear functions in coordinate geometry.)