Question

interactive digital version available: driving on i-90

1. a driver is traveling at a constant speed on interstate 90 outside chicago. if she traveled from point a to point b in 8 minutes, did she obey the speed limit of 55 miles per hour? explain your reasoning.

image of a map with points a and b

1. a traffic helicopter flew directly from point a to point b in 8 minutes. did the helicopter travel faster or slower than the driver? explain or show your reasoning.

Explanation:

To solve these problems, we need to determine the distance between Point A and Point B from the map (using the scale) and then calculate the speed of the driver and the helicopter. However, since the map's scale is not clearly visible in the provided image, we'll assume a common map scale or use the fact that in typical road maps, the distance between such points can be estimated. But for the sake of solving, let's assume the distance between Point A and Point B (by road for the driver) is, say, 6 miles (a common distance for such problems).

Problem 1: Driver's Speed

Step 1: Convert Time to Hours

The driver takes 8 minutes. Since 1 hour = 60 minutes, we convert 8 minutes to hours:
$ \text{Time (hours)} = \frac{8}{60} = \frac{2}{15} \text{ hours} $

Step 2: Calculate Speed

Speed is given by $ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $. Assuming the distance (by road) is 6 miles:
$ \text{Speed} = \frac{6}{\frac{2}{15}} = 6 \times \frac{15}{2} = 45 \text{ mph} $

Since 45 mph < 55 mph, the driver obeyed the speed limit.

Problem 2: Helicopter vs. Driver Speed

Step 1: Helicopter’s Path

The helicopter flies directly from A to B (straight line), which is shorter than the driver’s road distance (since roads are not straight). Let’s assume the straight-line distance is 5 miles (shorter than 6 miles).

Step 2: Helicopter’s Speed

Time is still 8 minutes = $ \frac{2}{15} $ hours.
$ \text{Speed (helicopter)} = \frac{5}{\frac{2}{15}} = 5 \times \frac{15}{2} = 37.5 \text{ mph} $ (Wait, no—wait, if the straight-line distance is shorter, but maybe my assumed distance is wrong. Wait, actually, if the driver’s distance is 6 miles (road) and the helicopter’s distance is shorter, but maybe the map scale is different. Alternatively, maybe the road distance is 8 miles? Let’s correct.

Wait, maybe the actual distance (from the map) is, say, 7 miles (road). Then driver’s speed: $ \frac{7}{\frac{8}{60}} = 7 \times \frac{60}{8} = 52.5 \text{ mph} $ (still under 55). But for the helicopter, the straight-line distance is shorter, so if the road distance is 7 miles, the straight-line distance might be 6 miles. Then helicopter’s speed: $ \frac{6}{\frac{8}{60}} = 6 \times \frac{60}{8} = 45 \text{ mph} $, which is slower than the driver’s 52.5 mph. But this depends on the map.

Alternatively, if the road distance is 8 miles (so driver’s speed: $ \frac{8}{\frac{8}{60}} = 60 \text{ mph} $, which would exceed 55, but that’s a different assumption).

Since the map is unclear, the key idea is:

• The driver travels the road distance (longer, due to roads), while the helicopter travels the straight-line distance (shorter).
• Thus, for the same time (8 minutes), the helicopter’s distance is shorter, so its speed is slower than the driver’s (if the driver’s speed is under 55, or faster if the driver’s speed is over). But based on typical problems, the driver’s speed is under 55, and the helicopter (flying straight) has a shorter distance, so slower.

Final Answers (Assuming Road Distance = 6 miles):

1. The driver’s speed is 45 mph, which is less than 55 mph, so she obeyed the speed limit.
2. The helicopter traveled slower than the driver because it flew a shorter (straight-line) distance in the same time.

(Note: The actual answer depends on the map’s scale and distance between A and B, which is not fully visible. The above is a typical solution approach.)

Answer:

Step 1: Helicopter’s Path

The helicopter flies directly from A to B (straight line), which is shorter than the driver’s road distance (since roads are not straight). Let’s assume the straight-line distance is 5 miles (shorter than 6 miles).

Step 2: Helicopter’s Speed

Time is still 8 minutes = $ \frac{2}{15} $ hours.
$ \text{Speed (helicopter)} = \frac{5}{\frac{2}{15}} = 5 \times \frac{15}{2} = 37.5 \text{ mph} $ (Wait, no—wait, if the straight-line distance is shorter, but maybe my assumed distance is wrong. Wait, actually, if the driver’s distance is 6 miles (road) and the helicopter’s distance is shorter, but maybe the map scale is different. Alternatively, maybe the road distance is 8 miles? Let’s correct.

Wait, maybe the actual distance (from the map) is, say, 7 miles (road). Then driver’s speed: $ \frac{7}{\frac{8}{60}} = 7 \times \frac{60}{8} = 52.5 \text{ mph} $ (still under 55). But for the helicopter, the straight-line distance is shorter, so if the road distance is 7 miles, the straight-line distance might be 6 miles. Then helicopter’s speed: $ \frac{6}{\frac{8}{60}} = 6 \times \frac{60}{8} = 45 \text{ mph} $, which is slower than the driver’s 52.5 mph. But this depends on the map.

Alternatively, if the road distance is 8 miles (so driver’s speed: $ \frac{8}{\frac{8}{60}} = 60 \text{ mph} $, which would exceed 55, but that’s a different assumption).

Since the map is unclear, the key idea is:

• The driver travels the road distance (longer, due to roads), while the helicopter travels the straight-line distance (shorter).
• Thus, for the same time (8 minutes), the helicopter’s distance is shorter, so its speed is slower than the driver’s (if the driver’s speed is under 55, or faster if the driver’s speed is over). But based on typical problems, the driver’s speed is under 55, and the helicopter (flying straight) has a shorter distance, so slower.

Final Answers (Assuming Road Distance = 6 miles):

1. The driver’s speed is 45 mph, which is less than 55 mph, so she obeyed the speed limit.
2. The helicopter traveled slower than the driver because it flew a shorter (straight-line) distance in the same time.

(Note: The actual answer depends on the map’s scale and distance between A and B, which is not fully visible. The above is a typical solution approach.)