Question
Three cars start traveling at the same time. The distance traveled by each is represented in three different ways, with y representing miles traveled and t representing time passed in hours.
Car 1:
y=t
Car 2:
t: 0, 0.2, 0.4, 0.6, 0.8, 1
y=t^2: 0, 0.04. 0.16, 0.36, 0.64, 1
Car 3: A graph starting a 0 and going to 5 is shown, there is a point starting a 0 and then going up to the top right corner and in the middle of that point y=2^t - 1 is shown
Compare the speeds of the 3 cars after 5 hours. In your answer, describe the function represented by each car.
Car 1:
y=t
Car 2:
t: 0, 0.2, 0.4, 0.6, 0.8, 1
y=t^2: 0, 0.04. 0.16, 0.36, 0.64, 1
Car 3: A graph starting a 0 and going to 5 is shown, there is a point starting a 0 and then going up to the top right corner and in the middle of that point y=2^t - 1 is shown
Compare the speeds of the 3 cars after 5 hours. In your answer, describe the function represented by each car.
Answers
GPT-4o mini
To compare the speeds of the three cars after 5 hours, we will analyze the function that represents each car's distance traveled in relation to time.
### Car 1:
- **Function**: \( y = t \)
- **Description**: This is a linear function where the distance traveled increases directly with time. The speed of Car 1 is constant at 1 mile per hour (since for every hour, the car travels 1 mile).
### Car 2:
- **Data points**:
- \( t: 0, 0.2, 0.4, 0.6, 0.8, 1 \)
- \( y: t^2: 0, 0.04, 0.16, 0.36, 0.64, 1 \)
- **Function**: \( y = t^2 \)
- **Description**: This is a quadratic function where the distance traveled increases with the square of the time. The speed of Car 2 is not constant; it increases over time. The speed can be found by taking the derivative of the function:
\[ v(t) = \frac{dy}{dt} = 2t \]
After 5 hours (\( t = 5 \)):
\[ v(5) = 2(5) = 10 \text{ miles per hour} \]
### Car 3:
- **Function**: \( y = 2^t - 1 \)
- **Description**: This is an exponential function, which means it increases very rapidly as time progresses. The speed is also not constant and increases as time goes on. To find the speed of Car 3, we also look at the derivative:
\[ v(t) = \frac{dy}{dt} = 2^t \ln(2) \]
After 5 hours:
\[ v(5) = 2^5 \ln(2) = 32 \ln(2) \approx 32(0.693) \approx 22.18 \text{ miles per hour} \]
### Summary After 5 Hours:
- **Car 1**: Speed = 1 mph
- **Car 2**: Speed = 10 mph
- **Car 3**: Speed = approximately 22.18 mph
### Conclusion:
After 5 hours, the speeds of the cars are as follows:
- Car 1 travels at a constant speed of 1 mph,
- Car 2 accelerates to a speed of 10 mph,
- Car 3, due to its exponential nature, reaches a higher speed of approximately 22.18 mph.
This shows that Car 3 has the highest speed after 5 hours, followed by Car 2, and then Car 1.
### Car 1:
- **Function**: \( y = t \)
- **Description**: This is a linear function where the distance traveled increases directly with time. The speed of Car 1 is constant at 1 mile per hour (since for every hour, the car travels 1 mile).
### Car 2:
- **Data points**:
- \( t: 0, 0.2, 0.4, 0.6, 0.8, 1 \)
- \( y: t^2: 0, 0.04, 0.16, 0.36, 0.64, 1 \)
- **Function**: \( y = t^2 \)
- **Description**: This is a quadratic function where the distance traveled increases with the square of the time. The speed of Car 2 is not constant; it increases over time. The speed can be found by taking the derivative of the function:
\[ v(t) = \frac{dy}{dt} = 2t \]
After 5 hours (\( t = 5 \)):
\[ v(5) = 2(5) = 10 \text{ miles per hour} \]
### Car 3:
- **Function**: \( y = 2^t - 1 \)
- **Description**: This is an exponential function, which means it increases very rapidly as time progresses. The speed is also not constant and increases as time goes on. To find the speed of Car 3, we also look at the derivative:
\[ v(t) = \frac{dy}{dt} = 2^t \ln(2) \]
After 5 hours:
\[ v(5) = 2^5 \ln(2) = 32 \ln(2) \approx 32(0.693) \approx 22.18 \text{ miles per hour} \]
### Summary After 5 Hours:
- **Car 1**: Speed = 1 mph
- **Car 2**: Speed = 10 mph
- **Car 3**: Speed = approximately 22.18 mph
### Conclusion:
After 5 hours, the speeds of the cars are as follows:
- Car 1 travels at a constant speed of 1 mph,
- Car 2 accelerates to a speed of 10 mph,
- Car 3, due to its exponential nature, reaches a higher speed of approximately 22.18 mph.
This shows that Car 3 has the highest speed after 5 hours, followed by Car 2, and then Car 1.