Answer :
To solve this problem, we need to break it down into parts and think about the mass intervals provided: [tex]$0 \leq x < 10$[/tex], [tex]$10 \leq x < 20$[/tex], [tex]$20 \leq x < 30$[/tex], and [tex]$30 \leq x < 40$[/tex].
### Step-by-Step Solution:
#### Part (a): Minimum number of dogs that could have a mass of more than 26 kg
1. Identify the relevant intervals:
- Any dog in the intervals [tex]$20 \leq x < 30$[/tex] or [tex]$30 \leq x < 40$[/tex] could potentially have a mass greater than 26 kg.
2. Dogs in the [tex]$30 \leq x < 40$[/tex] interval:
- All 5 dogs in this interval definitely have a mass greater than 26 kg.
3. Dogs in the [tex]$20 \leq x < 30$[/tex] interval:
- For the minimum possible number, only one dog in this interval needs to have a mass greater than 26 kg. The remaining dogs would then have to be 26 kg or less.
- Since there are 11 dogs in this interval, at least 10 dogs could have a mass less than or equal to 26 kg.
Therefore, if only 1 dog in that interval is over 26 kg, we add this to the 5 dogs from the [tex]$30 \leq x < 40$[/tex] interval.
4. Sum these counts:
- Minimum number of dogs over 26 kg is [tex]\(5 + 1 = 6\)[/tex].
#### Part (b): Maximum number of dogs that could have a mass of more than 26 kg
1. Identify the dogs that definitely have a mass more than 26 kg:
- All 5 dogs in the [tex]$30 \leq x < 40$[/tex] interval.
2. Make an assumption for maximum count:
- Assume all dogs in the [tex]$20 \leq x < 30$[/tex] interval could potentially be over 26 kg.
- Since this interval ranges from 20 to below 30 kg, it is possible that all 11 dogs in this interval have a mass between 27 and 29 kg (inclusive).
3. Sum these counts:
- Maximum number of dogs over 26 kg is the sum of all dogs in the [tex]$20 \leq x < 30$[/tex] interval plus the dogs in the [tex]$30 \leq x < 40$[/tex] interval.
- This gives [tex]\(11 + 5 = 16\)[/tex].
So, the results are:
- Minimum number of dogs with a mass more than 26 kg: 15
- Maximum number of dogs with a mass more than 26 kg: 16
### Step-by-Step Solution:
#### Part (a): Minimum number of dogs that could have a mass of more than 26 kg
1. Identify the relevant intervals:
- Any dog in the intervals [tex]$20 \leq x < 30$[/tex] or [tex]$30 \leq x < 40$[/tex] could potentially have a mass greater than 26 kg.
2. Dogs in the [tex]$30 \leq x < 40$[/tex] interval:
- All 5 dogs in this interval definitely have a mass greater than 26 kg.
3. Dogs in the [tex]$20 \leq x < 30$[/tex] interval:
- For the minimum possible number, only one dog in this interval needs to have a mass greater than 26 kg. The remaining dogs would then have to be 26 kg or less.
- Since there are 11 dogs in this interval, at least 10 dogs could have a mass less than or equal to 26 kg.
Therefore, if only 1 dog in that interval is over 26 kg, we add this to the 5 dogs from the [tex]$30 \leq x < 40$[/tex] interval.
4. Sum these counts:
- Minimum number of dogs over 26 kg is [tex]\(5 + 1 = 6\)[/tex].
#### Part (b): Maximum number of dogs that could have a mass of more than 26 kg
1. Identify the dogs that definitely have a mass more than 26 kg:
- All 5 dogs in the [tex]$30 \leq x < 40$[/tex] interval.
2. Make an assumption for maximum count:
- Assume all dogs in the [tex]$20 \leq x < 30$[/tex] interval could potentially be over 26 kg.
- Since this interval ranges from 20 to below 30 kg, it is possible that all 11 dogs in this interval have a mass between 27 and 29 kg (inclusive).
3. Sum these counts:
- Maximum number of dogs over 26 kg is the sum of all dogs in the [tex]$20 \leq x < 30$[/tex] interval plus the dogs in the [tex]$30 \leq x < 40$[/tex] interval.
- This gives [tex]\(11 + 5 = 16\)[/tex].
So, the results are:
- Minimum number of dogs with a mass more than 26 kg: 15
- Maximum number of dogs with a mass more than 26 kg: 16
