Johanna and Carl were sitting at tea one afternoon when suddenly Bob hopped over to their table.
“Well, Johanna,” he said, “Have you considered my offer?”
“Yes, I have, and the answer is still no,” said Johanna. “I told you I have no interest in another race. Even though I technically beat you, everyone knows that you run twice as fast as me. I see no point in a rematch.”
“I said no, Bob.”
Bob turned to her companion.
“Carl, you talk to her. Get her to change her mind.”
Carl merely looked at Bob and then started licking his paw. Johanna took another sip of tea. Bob’s ears drooped as he turned and walked away.
“Honestly,” said Johanna, “I don’t know what to do to discourage him. Every few days, he pesters me about another race. What should I do?”
Carl thought for a minute while he scratched himself.
“Let me look into it,” he said. “I will gather some data, crunch the numbers, and get back to you.”
He and Joanna finished their tea.
Carl spent the next few days watching Johanna and Bob run. He made a table of values showing Johanna’s velocity and wrote a function that represented Bob’s velocity. Then he showed the data to Bob, pointing out to him the different velocities. Bob grew excited when he saw the data and told Carl that he would give Johanna a twenty-minute head start on the race. Carl said he would talk to Johanna.
The next day, Carl met with Johanna and showed her the data.
“Bob said he would give you a twenty-minute head start on the race, seeing how it takes you forty minutes to run the course. I recommend you accept his challenge.”
“Accept!” Johanna said. “Are you crazy? Bob won’t fool around like he did last time; he’ll turn on the steam and blow right past me.”
“Trust me on this,” said Carl. “Accept the challenge.”
Word swept through the countryside that Johanna and Bob were going to run a rematch race. The odds were calculated and bets were placed in favor of Bob winning. Spectators lined the course the day of the race. Excitement was high. At the signal, Johanna took off running. Bob made a great show of waiting patiently for twenty minutes to transpire; then he started running after Johanna. He was confident that he would win the race, having used a left Riemann sum to approximate Johanna’s velocity at 152 meters per minute. Based on Carl’s function, he calculated that his velocity would be 350 meters per minute. Hence, even with a head start, Bob figured he would overtake Johanna. He almost made it.
As expected, it took Johanna forty minutes to run the course, but it took Bob 40.18557 minutes. Johanna won the race a second time. Bob never again mentioned a rematch.
That evening over dinner, Johanna thanked Carl for advising her to run the race.
“How did you know I would win?’ she asked.
“It was easy,” said Carl, “once I showed the data to Bob. He calculated your velocity at 152 meters per minute using a left Riemann sum and thought he could outrun you, even after a twenty-minute delay. However, I calculated your velocity using a right Riemann sum, which makes it 190 meters per minute.”
“How did you know Bob used a left Riemann sum?”
“If he had known your velocity was greater than his, he never would have agreed to the race.”
“Well, I am grateful for all your help,” said Johanna, “including this dinner. It’s absolutely delicious! But, Carl, isn’t all this a little expensive? I mean, you being a cat and all, how can you afford it?”
Carl licked his paw.
“Who do you think bet on you?”