zorba and English mastiff with a mass of 155 kg jumps forward horizontally at a speed of 6 m/s into a boat that is floating at rest. after the jump, the boat and zobra move with a velocity of 2.2 m/s forward. calculate the boats mass.
- In a distribution with a standard deviation of eight, an individual score of 42 corresponds to a Z score of -0.5. What is the mean of this distribution?
- The distribution of SAT scores is normal with u=500 and o= 100a. What SAT score, X value, separate the top 15% of the distribution from the rest?b. What SAT score, X value, separates the top 10% of the distribution from the rest?c. What SAT score, X value, separates the top 2% of the distribution from the rest?
The Correct Answer and Explanation is :
1. Calculating the Boat’s Mass
We will use the principle of conservation of momentum, as no external horizontal forces are acting on the system (Zorba and the boat). According to this principle, the momentum before the jump equals the momentum after the jump.
Before the jump:
- Zorba’s mass ( m_z = 155 \, \text{kg} )
- Zorba’s velocity ( v_z = 6 \, \text{m/s} )
- Boat’s mass ( m_b ) (which we need to find)
- Boat’s velocity ( v_b = 0 \, \text{m/s} ) (since the boat is at rest)
Momentum before the jump:
[
\text{Momentum before} = (m_z \times v_z) + (m_b \times v_b) = (155 \, \text{kg} \times 6 \, \text{m/s}) + (m_b \times 0) = 930 \, \text{kg} \cdot \text{m/s}
]
After the jump:
- Both Zorba and the boat move with the same velocity, ( v_f = 2.2 \, \text{m/s} )
Momentum after the jump:
[
\text{Momentum after} = (m_z + m_b) \times v_f = (155 + m_b) \times 2.2
]
Since momentum is conserved, we set the momentum before equal to the momentum after:
[
930 = (155 + m_b) \times 2.2
]
Solving for ( m_b ):
[
930 = 2.2 \times (155 + m_b)
]
[
930 = 341 + 2.2 \times m_b
]
[
930 – 341 = 2.2 \times m_b
]
[
589 = 2.2 \times m_b
]
[
m_b = \frac{589}{2.2} = 267.7 \, \text{kg}
]
Thus, the boat’s mass is approximately 267.7 kg.
2. Finding the Mean of a Distribution Using Z-Score
The formula for the Z-score is:
[
Z = \frac{X – \mu}{\sigma}
]
Where:
- ( Z ) is the Z-score,
- ( X ) is the score,
- ( \mu ) is the mean,
- ( \sigma ) is the standard deviation.
Given:
- ( Z = -0.5 )
- ( X = 42 )
- ( \sigma = 8 )
We substitute into the Z-score formula and solve for ( \mu ):
[
-0.5 = \frac{42 – \mu}{8}
]
Multiplying both sides by 8:
[
-4 = 42 – \mu
]
Solving for ( \mu ):
[
\mu = 42 + 4 = 46
]
Thus, the mean of the distribution is 46.
3. SAT Score Distribution Questions
We are given that the SAT scores follow a normal distribution with:
- Mean ( \mu = 500 )
- Standard deviation ( \sigma = 100 )
To find the SAT scores that separate different percentages of the distribution, we will use the Z-score formula.
a. SAT score that separates the top 15%:
The Z-score corresponding to the top 15% (the 85th percentile) can be found using a Z-table or calculator. The Z-score for the 85th percentile is approximately ( Z = 1.036 ).
Now, we use the Z-score formula:
[
Z = \frac{X – \mu}{\sigma}
]
Substitute the values:
[
1.036 = \frac{X – 500}{100}
]
Solving for ( X ):
[
X – 500 = 1.036 \times 100
]
[
X – 500 = 103.6
]
[
X = 500 + 103.6 = 603.6
]
So, the SAT score that separates the top 15% is approximately 604.
b. SAT score that separates the top 10%:
The Z-score for the 90th percentile is approximately ( Z = 1.282 ).
[
1.282 = \frac{X – 500}{100}
]
Solving for ( X ):
[
X – 500 = 1.282 \times 100
]
[
X – 500 = 128.2
]
[
X = 500 + 128.2 = 628.2
]
So, the SAT score that separates the top 10% is approximately 628.
c. SAT score that separates the top 2%:
The Z-score for the 98th percentile is approximately ( Z = 2.054 ).
[
2.054 = \frac{X – 500}{100}
]
Solving for ( X ):
[
X – 500 = 2.054 \times 100
]
[
X – 500 = 205.4
]
[
X = 500 + 205.4 = 705.4
]
So, the SAT score that separates the top 2% is approximately 705.
Explanation
The first question involves applying the principle of conservation of momentum. When Zorba jumps into the boat, the system’s total momentum remains unchanged, allowing us to calculate the boat’s mass. In the second question, we use the formula for Z-scores to find the mean of a distribution when the Z-score and an individual value are known. Finally, for the SAT scores, we used Z-scores corresponding to specific percentiles (85th, 90th, and 98th) to find the scores that separate the top 15%, 10%, and 2% of the distribution. These calculations are essential for understanding how data distributions work and how standard deviation and percentiles can help us interpret data.
Let me generate a visualization for these questions.
Here is an illustration that visualizes the conservation of momentum in the scenario with the dog (Zorba) and the boat. It shows the dog jumping into the boat and both moving with a common velocity afterward. This can help to further understand the momentum conservation concept from the first question.
Let me know if you need anything else!