Richard and Gloria ran a 10-kilometer race. Gloria gave Richard a head start by letting him begin 2.2 kilometers in front of the start line. The distances of Richard and Gloria past the start line as the race progressed can be represented by linear functions and are shown in the table of values and graph below. Whose time was faster, and by how much?

Question

Richard’s Distance Past the Start Line
Time (minutes)
5
10
15
20
25
Distance (kilometers)
2.8
3.4
4.0
4.6
5.2

A graph titled Gloria's Distance Past the Start Line has time (minutes) on the x-axis and Distance (kilometers) on the y-axis. A line goes through points (5, 0.8), (10, 1.6), (15, 2.4), (20, 3.2) and (25, 4).
Gloria’s time was 0.5 minutes faster.
Richard’s time was 0.5 minutes faster.
Gloria’s time was 2.5 minutes faster.
Richard’s time was 2.5 minutes faster.

Answer 1

To determine whose time was faster and by how much, we first need to analyze the distances Richard and Gloria traveled based on the table provided for Richard and the graph for Gloria.

Richard's Distance:

Richard's distances at different times are:

• At 5 minutes: 2.8 km
• At 10 minutes: 3.4 km
• At 15 minutes: 4.0 km
• At 20 minutes: 4.6 km
• At 25 minutes: 5.2 km

Gloria's Distance:

From the graph, we can find Gloria's distances at the same times:

• At 5 minutes: 0.8 km
• At 10 minutes: 1.6 km
• At 15 minutes: 2.4 km
• At 20 minutes: 3.2 km
• At 25 minutes: 4.0 km

Race Progression:

Both Richard and Gloria are running a total of 10 km, but Richard started at 2.2 km past the start line.

One important point to calculate is when each of them reaches the finish line (10 km).

Richard:

Richard's distance at 25 minutes was 5.2 km. To find out when he will reach 10 km, we need to look at his rate of distance covered.

From the data:

• In 25 minutes, Richard covers 5.2 km.
• His rate can be calculated as: $$\text{Rate}=\frac{5.2\text{ km}}{25\text{ minutes}}=0.208\text{ km/min}$$ To find the total distance Richard needs to cover, we can find the distance he has left: $$\text{Distance remaining}=10\text{ km}-2.2\text{ km}-5.2\text{ km}=2.6\text{ km}$$ Now we can calculate how long it will take Richard to cover the remaining 2.6 km: $$\text{Time remaining}=\frac{2.6\text{ km}}{0.208\text{ km/min}}\approx 12.5\text{ minutes}$$ Therefore, Richard’s total time to finish the race will be: $$\text{Total time for Richard}=25\text{ minutes}+12.5\text{ minutes}=37.5\text{ minutes}$$

Gloria:

Next, we calculate the time for Gloria. From the recorded distances in the graph for 25 minutes, Gloria covered 4 km. We can similarly find her rate: $$\text{Rate}=\frac{4\text{ km}}{25\text{ minutes}}=0.16\text{ km/min}$$ Now, to find the total time required for Gloria to reach 10 km: $$\text{Distance remaining}=10\text{ km}-4.0\text{ km}=6.0\text{ km}$$ The time needed for Gloria to cover the remaining distance is: $$\text{Time remaining}=\frac{6.0\text{ km}}{0.16\text{ km/min}}=37.5\text{ minutes}$$ Thus, Gloria will finish the race in: $$\text{Total time for Gloria}=25\text{ minutes}+37.5\text{ minutes}=62.5\text{ minutes}$$

Conclusion:

Comparing both times:

• Richard: 37.5 minutes
• Gloria: 62.5 minutes

Thus, Richard’s time is faster than Gloria's time.

$$\text{Difference}=62.5-37.5=25\text{ minutes}$$

Correct Answer: Richard's time was 25 minutes faster.