The minute hand of a clock is 7 inches long. How far does the tip move in 16 minutes?
a.56/15pi
b.8/105pi
c.4/105pi
d.28/15pi
in 60 minutes, the minute hand moves 2π radians
in 16 minutes it moves 16(2π/60) or 8π/15 radians
arc = rØ = 7(8π/15) = 56π/15
looks like a)
To find out how far the tip of the minute hand moves in 16 minutes, we need to find the length of the arc that the tip traces.
The minute hand of a clock makes a complete revolution every 60 minutes. This means that in 60 minutes, it moves around the entire circumference of a circle with a radius equal to its length.
Given that the length of the minute hand is 7 inches, the radius of the circle is also 7 inches.
To find the length of the arc traced by the tip in 16 minutes, we can use the formula for the length of a circular arc:
Arc Length = (angle in radians) * (radius)
In this case, the angle in radians can be found by dividing the time in minutes (16) by the time it takes for the minute hand to make a complete revolution (60 minutes), and then multiplying by 2π (since there are 2π radians in a full circle).
angle in radians = (16 / 60) * 2π
Simplifying this expression gives:
angle in radians = (4/15)π
Finally, we can substitute this value into the arc length formula:
Arc Length = (4/15)π * 7
Simplifying further, we get:
Arc Length = 28/15π
Therefore, the correct answer is option d. 28/15π.
To find how far the tip of the minute hand moves in 16 minutes, we need to calculate the arc length of the circle that the tip covers in that time.
The formula to find the arc length is:
Arc Length = (Angle in radians) x (Radius)
First, let's find the angle covered by the minute hand. The minute hand completes a full revolution (360 degrees) in 60 minutes, so in 16 minutes it will cover:
Angle = (16/60) x 360 = 96 degrees
To convert the angle to radians, we use the formula:
Radians = (Degrees x Pi) / 180
Radians = (96 x Pi) / 180 = 16Pi / 30
Now, we can substitute the angle and the radius into the arc length formula:
Arc Length = (16Pi / 30) x 7
Simplifying, we get:
Arc Length = (112Pi) / 30
Now, let's simplify the expression:
Arc Length = (56Pi) / 15
Therefore, the tip of the minute hand moves (56Pi / 15) inches in 16 minutes.
Therefore, the answer is a. 56/15Pi.