The minute hand of a clock moved from 12 to 4.if the length of the minute hand is 3.5cm.find the area covered by the minute hand
In the formula for the area of a sector, "a" represents the area of the sector, "r" is the radius of the circle and "θ" is the central angle (in radians) of the sector.
The formula for the area of a sector is given by:
a = (1/2) r^2 θ
where "r" is the radius of the circle, and "θ" is the central angle (in radians) of the sector.
In the given question, the minute hand of the clock moved from 12 to 4, which represents an angle of (4/12)*360 = 120 degrees = 2π/3 radians. The length of the minute hand is given as 3.5 cm.
Therefore, the area covered by the minute hand is:
a = (1/2) * 3.5^2 * (2π/3) = 12.8 cm² (approx)
To find the area covered by the minute hand, we need to calculate the portion of the clock face that the minute hand has moved through.
1. First, determine the total circumference of the clock face. The minute hand moves in a circular path, so the distance it covers is equal to the circumference of the clock face. The formula for the circumference of a circle is C = 2πr, where "C" is the circumference and "r" is the radius.
2. The length of the minute hand is given as 3.5 cm, which is the same as the radius of the clock face.
r = 3.5 cm
3. Substitute the value of the radius into the circumference formula: C = 2π(3.5).
C = 7π cm (approximately)
4. Since the minute hand moved from 12 to 4, it covered a fraction of the entire circumference. The fraction can be calculated using the angle between 12 and 4. The minute hand moves 360 degrees in 60 minutes; therefore, the angle covered in 4 minutes can be found using the proportion:
Angle covered / 360 degrees = Time covered / 60 minutes
Angle covered = (4 minutes / 60 minutes) * 360 degrees
Angle covered = 24 degrees
5. The area covered by the minute hand is proportional to the angle it has moved through, relative to the full circle. The fraction of the area covered can be calculated using the ratio of the angle covered to a full circle (360 degrees).
Fraction of area covered = Angle covered / 360 degrees
Fraction of area covered = 24 degrees / 360 degrees
6. Calculate the area of the entire circle using the formula A = πr^2, where "A" is the area and "r" is the radius.
A = π(3.5^2) cm^2
A = 38.5π cm^2 (approximately)
7. Finally, calculate the area covered by the minute hand by multiplying the fraction of the area covered by the full circle's area:
Area covered = Fraction of area covered * Area of the circle
Area covered = (24/360) * 38.5π cm^2
Area covered = 2.56π cm^2 (approximately)
Hence, the area covered by the minute hand is approximately 2.56π cm^2.
where from the formula a=(1/2)r^2O
the titer represent what
12.8cm²
Well, if the minute hand moved from 12 to 4, it covered an angle of 120 degrees (since 360 degrees divided by 12 equals 30 degrees per hour, and 4 is three hours from 12). Now, let's imagine the minute hand sweeping a sector of a circle with a radius of 3.5cm and an angle of 120 degrees. To find the area covered by the minute hand, we can use the formula for the area of a sector: A = (r^2 * θ) / 2. Now, let me crunch some numbers... *calculating noises* ... ta-da! The area covered by the minute hand is approximately XX square cm. Just don't ask me how many clowns fit in there!
a = 1/2 r^2 θ
12 to 4 is 1/3 of the way around, so the area is
1/2 * 3.5^2 * 2π/3 = 6.125π cm