The minute hand on a clock is 19mm long.What area does it pass over in 30 seconds.
the minute hand rotates 360° in 60 minutes
or
6° per minute
or
3° in 30 seconds.
area swept by minute hand in one rotation
= π(19^2) = 361π mm^2
so by ratios:
3/360 = area/361π
area = 361π(1/120) = 361π/120 mm^2
= appr 9.45 mm^2
check my arithmetic
It is very difficult to understand explain or break it down further.
To find the area that the minute hand passes over in 30 seconds, we can break down the problem into smaller steps:
Step 1: Determine the distance covered by the minute hand in 30 seconds.
The minute hand takes 60 minutes to complete a full revolution around the clock, which is equivalent to 360 degrees. Therefore, in 1 minute (60 seconds), the minute hand covers a distance of 360/60 = 6 degrees.
In 30 seconds, the minute hand covers half of that distance, which is 6/2 = 3 degrees.
Step 2: Convert the distance covered by the minute hand in degrees into radians.
To calculate the area, we need to work with radians rather than degrees. One full revolution (360 degrees) is equal to 2π radians.
Therefore, the distance covered by the minute hand in radians is (3/360) × 2π = π/60.
Step 3: Calculate the length of the arc covered by the minute hand.
The length of an arc in a circle is given by the formula: arc length = radius × angle (in radians).
In this case, the radius is given as 19 mm, and the angle is π/60 radians. So, the length of the arc covered by the minute hand in 30 seconds is (19 mm) × (π/60) ≈ 0.316 mm (rounded to three decimal places).
Step 4: Calculate the area covered by the arc.
To calculate the area, we need to multiply the length of the arc by the width of the minute hand.
Assuming the width of the minute hand is negligible compared to the length, we can approximate the area as a rectangle with length 0.316 mm and width 1 mm (since the width is not given, we'll assume 1 mm).
Therefore, the area covered by the minute hand in 30 seconds is approximately 0.316 mm².
In summary, the area that the minute hand passes over in 30 seconds is approximately 0.316 mm².