Answer :
Sure, let's work through the problem step by step to determine the work done by the boat on Matthew.
1. Given Data:
- Force ([tex]\( F \)[/tex]): [tex]\( F = 250 \)[/tex] newtons.
- Angle ([tex]\( \theta \)[/tex]): [tex]\( \theta = 8.0^\circ \)[/tex].
- Distance ([tex]\( d \)[/tex]): Not explicitly given, so we assume a reasonable value for calculation purposes.
2. Work Formula:
The work [tex]\( W \)[/tex] done by a force applied at an angle is given by:
[tex]\[ W = F \cdot d \cdot \cos(\theta) \][/tex]
where:
- [tex]\( F \)[/tex] is the force applied,
- [tex]\( d \)[/tex] is the distance over which the force is applied,
- [tex]\( \theta \)[/tex] is the angle between the direction of the force and the direction of motion,
- [tex]\( \cos(\theta) \)[/tex] is the cosine of the angle.
3. Convert the Angle to Radians:
To use the cosine function, we first need to convert the angle from degrees to radians:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \left(\frac{\pi}{180}\right) \][/tex]
[tex]\[ \theta_{\text{radians}} = 8.0^\circ \times \left(\frac{\pi}{180}\right) \approx 0.14 \, \text{radians} \][/tex]
4. Calculate the Cosine of the Angle:
[tex]\[ \cos(8.0^\circ) \approx \cos(0.14 \, \text{radians}) \approx 0.99 \][/tex]
5. Estimating Distance [tex]\( d \)[/tex]:
Assume the boat pulls Matthew over a distance of [tex]\( 100 \)[/tex] meters.
6. Calculate Work Done:
Substituting the values into the work formula:
[tex]\[ W = 250 \, \text{N} \times 100 \, \text{m} \times \cos(8.0^\circ) \][/tex]
[tex]\[ W \approx 250 \, \text{N} \times 100 \, \text{m} \times 0.99 \][/tex]
[tex]\[ W \approx 25000 \times 0.99 \][/tex]
[tex]\[ W \approx 24750 \, \text{joules} \][/tex]
7. Rounding to Closest Answer:
Looking at the options provided:
- A. [tex]\( 1.0 \times 10^4 \)[/tex] joules
- B. [tex]\( 1.2 \times 10^4 \)[/tex] joules
- C. [tex]\( 1.4 \times 10^4 \)[/tex] joules
- D. [tex]\( 1.7 \times 10^4 \)[/tex] joules
The work done [tex]\( 24750 \)[/tex] joules is closest to option D, which is [tex]\( 1.7 \times 10^4 \)[/tex] joules.
So, the correct answer is:
D. [tex]\( 1.7 \times 10^4 \)[/tex] joules
1. Given Data:
- Force ([tex]\( F \)[/tex]): [tex]\( F = 250 \)[/tex] newtons.
- Angle ([tex]\( \theta \)[/tex]): [tex]\( \theta = 8.0^\circ \)[/tex].
- Distance ([tex]\( d \)[/tex]): Not explicitly given, so we assume a reasonable value for calculation purposes.
2. Work Formula:
The work [tex]\( W \)[/tex] done by a force applied at an angle is given by:
[tex]\[ W = F \cdot d \cdot \cos(\theta) \][/tex]
where:
- [tex]\( F \)[/tex] is the force applied,
- [tex]\( d \)[/tex] is the distance over which the force is applied,
- [tex]\( \theta \)[/tex] is the angle between the direction of the force and the direction of motion,
- [tex]\( \cos(\theta) \)[/tex] is the cosine of the angle.
3. Convert the Angle to Radians:
To use the cosine function, we first need to convert the angle from degrees to radians:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \left(\frac{\pi}{180}\right) \][/tex]
[tex]\[ \theta_{\text{radians}} = 8.0^\circ \times \left(\frac{\pi}{180}\right) \approx 0.14 \, \text{radians} \][/tex]
4. Calculate the Cosine of the Angle:
[tex]\[ \cos(8.0^\circ) \approx \cos(0.14 \, \text{radians}) \approx 0.99 \][/tex]
5. Estimating Distance [tex]\( d \)[/tex]:
Assume the boat pulls Matthew over a distance of [tex]\( 100 \)[/tex] meters.
6. Calculate Work Done:
Substituting the values into the work formula:
[tex]\[ W = 250 \, \text{N} \times 100 \, \text{m} \times \cos(8.0^\circ) \][/tex]
[tex]\[ W \approx 250 \, \text{N} \times 100 \, \text{m} \times 0.99 \][/tex]
[tex]\[ W \approx 25000 \times 0.99 \][/tex]
[tex]\[ W \approx 24750 \, \text{joules} \][/tex]
7. Rounding to Closest Answer:
Looking at the options provided:
- A. [tex]\( 1.0 \times 10^4 \)[/tex] joules
- B. [tex]\( 1.2 \times 10^4 \)[/tex] joules
- C. [tex]\( 1.4 \times 10^4 \)[/tex] joules
- D. [tex]\( 1.7 \times 10^4 \)[/tex] joules
The work done [tex]\( 24750 \)[/tex] joules is closest to option D, which is [tex]\( 1.7 \times 10^4 \)[/tex] joules.
So, the correct answer is:
D. [tex]\( 1.7 \times 10^4 \)[/tex] joules
