Let's solve the problem step by step to find the work done by a 20 Newton force applied at an angle of [tex]\( 45.0^\circ \)[/tex] to move a box a distance of 40 meters.
Step 1: Understanding the Formula for Work Done
The formula to calculate the work done [tex]\( W \)[/tex] when a force [tex]\( F \)[/tex] is applied at an angle [tex]\( \theta \)[/tex] to the direction of movement over a distance [tex]\( d \)[/tex] is:
[tex]\[ W = F \cdot d \cdot \cos(\theta) \][/tex]
where:
- [tex]\( W \)[/tex] is the work done,
- [tex]\( F \)[/tex] is the force applied,
- [tex]\( d \)[/tex] is the distance moved,
- [tex]\( \theta \)[/tex] is the angle between the force and the direction of movement, and
- [tex]\( \cos(\theta) \)[/tex] is the cosine of the angle.
Step 2: Given Values
We have the following values:
- Force, [tex]\( F = 20 \)[/tex] Newtons,
- Angle, [tex]\( \theta = 45.0^\circ \)[/tex],
- Distance, [tex]\( d = 40 \)[/tex] meters.
Step 3: Converting the Angle to Radians
Angles in trigonometric functions in calculus are often expressed in radians. To convert degrees to radians, we use the formula:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
For [tex]\( 45.0^\circ \)[/tex], the conversion is:
[tex]\[ \theta_{\text{radians}} = 45.0 \times \frac{\pi}{180} = \frac{\pi}{4} \][/tex]
Step 4: Calculating the Cosine of the Angle
Using the angle in radians:
[tex]\[ \cos\left(\frac{\pi}{4}\right) = \cos(45.0^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
Step 5: Plugging the Values into the Formula
Now, substitute the values into the work done formula:
[tex]\[ W = 20 \cdot 40 \cdot \cos(45.0^\circ) \][/tex]
[tex]\[ W = 20 \cdot 40 \cdot \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ W = 20 \cdot 40 \cdot 0.707 \][/tex]
[tex]\[ W = 800 \cdot 0.707 \][/tex]
After performing the multiplication:
[tex]\[ W \approx 565.7 \, \text{J} \][/tex]
Step 6: Expressing the Work Done in Scientific Notation
The result in scientific notation:
[tex]\[ W \approx 5.7 \times 10^2 \, \text{J} \][/tex]
Step 7: Matching the Options
The closest option to our calculated value is:
C. [tex]\( 5.6 \times 10^2 \)[/tex]
Thus, the correct answer is:
C. [tex]\( 5.6 \times 10^2 \)[/tex] joules
