To find the size of the angle that the minute hand makes at the center when it covers a length of 14 cm, we will approach this step-by-step for each unit of measure: radians, degrees, and grades.
### Step 1: Calculate the angle in Radians
1. Given:
- Length of the minute hand (radius) = 7 cm
- Arc length covered by the minute hand = 14 cm
2. Formula:
[tex]\[ \text{Angle in radians} (\theta) = \frac{\text{Arc length}}{\text{Radius}} \][/tex]
3. Calculation:
[tex]\[ \theta = \frac{14 \text{ cm}}{7 \text{ cm}} = 2 \text{ radians} \][/tex]
So, the angle in radians is [tex]\( 2 \text{ radians} \)[/tex].
### Step 2: Convert the angle to Degrees
1. Formula:
[tex]\[ \text{Angle in degrees} = \text{Angle in radians} \times \left(\frac{180}{\pi}\right) \][/tex]
2. Calculation:
[tex]\[ \text{Angle in degrees} = 2 \times \left(\frac{180}{\pi}\right) \approx 2 \times 57.2958 \approx 114.5916 \text{ degrees} \][/tex]
So, the angle in degrees is approximately [tex]\( 114.5916 \text{ degrees} \)[/tex].
### Step 3: Convert the angle to Grades
1. Formula:
[tex]\[ \text{Angle in grades} = \text{Angle in degrees} \times \left(\frac{10}{9}\right) \][/tex]
2. Calculation:
[tex]\[ \text{Angle in grades} = 114.5916 \times \left(\frac{10}{9}\right) \approx 127.324 \text{ grades} \][/tex]
So, the angle in grades is approximately [tex]\( 127.324 \text{ grades} \)[/tex].
### Summary
(i) The angle in radians is [tex]\( 2 \text{ radians} \)[/tex].
(ii) The angle in degrees is approximately [tex]\( 114.5916 \text{ degrees} \)[/tex].
(iii) The angle in grades is approximately [tex]\( 127.324 \text{ grades} \)[/tex].
