Let's solve each question step by step:
Question 2:
- Peter runs at a pace of 10 minutes per mile.
- He starts running at 7:00 am.
- Therefore, in 15 minutes (by 7:15 am), he has covered \( \frac{15}{10} = 1.5 \) miles.
- Alice runs at a pace of 5 minutes per mile and starts at 7:15 am.
Let's denote the time from 7:15 am as \( t \). In \( t \) minutes:
- Alice runs \( \frac{t}{5} \) miles.
- Peter runs \( 1.5 + \frac{t}{10} \) miles.
To find when Alice catches up with Peter: \[ \frac{t}{5} = 1.5 + \frac{t}{10} \]
Multiplying the entire equation by 10 to eliminate the fractions: \[ 2t = 15 + t \] \[ 2t - t = 15 \] \[ t = 15 \]
Since Alice starts running at 7:15 am, she catches up with Peter at: \[ 7:15 am + 15 minutes = 7:30 am \]
Response for Question 2: 7:30 am
Question 3: The cube has 6 faces, each divided into 4 squares, so:
- Total number of squares = \( 6 \times 4 = 24 \)
To satisfy the condition that each face of the cube must have at least one square of each color (red and green), we can consider the following strategy:
- If we have 1 red square and 3 green squares on each face, we can maximize green squares while still meeting the requirement.
Calculating the total number of green squares: \[ 3\text{ green squares/face} \times 6\text{ faces} = 18\text{ green squares} \]
Thus, the greatest number of green squares that can be painted, while satisfying the color requirement, is: Response for Question 3: 18
Question 4: The available juices are:
- Carrot
- Celery
- Apple
- Pear
We need to find combinations of 2 juices out of these 4. This is a combinations problem, expressed as: \[ \text{Combination} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of options, and \( r \) is the number of selections.
For this case: \[ n = 4 \quad \text{and} \quad r = 2 \] \[ \text{Combinations} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
Response for Question 4: 6
Final Responses:
- Question 2: 7:30 am
- Question 3: 18
- Question 4: 6