The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 5 minutes? If necessary, round the answer to two decimal places.
1.7
To calculate how far the tip of the minute hand moves in 5 minutes, we need to find the arc length covered by the minute hand.
The minute hand of a clock moves in a circular motion with the center at the clock's center. The length of the minute hand represents the radius of this circle, which is given as 3 inches.
To find the arc length covered in 5 minutes, we need to calculate the proportion of the circumference covered by the minute hand in 5 minutes.
The circumference of a circle is given by: 2πr, where r is the radius.
In this case, the radius is 3 inches. So, the circumference of the circle is: 2π × 3 = 6π inches.
To find the proportion covered in 5 minutes, we divide 5 minutes by 60 minutes (the total number of minutes in an hour).
Proportion covered = 5 minutes / 60 minutes = 1/12.
To calculate the arc length covered, we multiply the proportion covered by the circumference of the circle.
Arc length covered = (1/12) × 6π inches.
Now, let's calculate the arc length:
Arc length covered = (1/12) × 6π = (1/2)π ≈ 1.57 inches.
Therefore, the tip of the minute hand moves approximately 1.57 inches in 5 minutes.
To determine how far the tip of the minute hand moves in 5 minutes, we need to calculate the length of the arc it travels.
The minute hand completes a full revolution, or 360 degrees, every 60 minutes. Therefore, in 1 minute, it will move 360 degrees / 60 minutes = 6 degrees.
To find the arc length for 5 minutes, we can calculate the portion of the circumference of the clock covered by the angle.
The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius.
Given that the minute hand is 3 inches long, the radius of the clock will be 3 inches.
Using the formula, we can calculate the circumference as C = 2 * π * 3 = 6π inches.
Now we need to find the portion of the circumference covered by an angle of 5 minutes.
To calculate this, we use the formula:
Arc length = (angle ÷ 360 degrees) * Circumference of the circle
Since we found earlier that the minute hand moves 6 degrees in 1 minute, the angle for 5 minutes would be 5 minutes * 6 degrees/minute = 30 degrees.
Plugging in these values into the formula gives us:
Arc length = (30 degrees ÷ 360 degrees) * 6π inches
Simplifying the expression:
Arc length = (1/12) * 6π inches
Arc length = 0.5π inches
Now we can calculate the approximate value in decimal form by substituting the value of π as 3.14:
Arc length ≈ 0.5 * 3.14 inches = 1.57 inches
Therefore, the tip of the minute hand moves approximately 1.57 inches in 5 minutes.
5 minutes is 1/12 of an hour.
The hand makes one complete circle in an hour
So, it moves 1/12 of the whole circumference in 5 minutes.
I assume you know how to find the circumference of a circle of radius 3.
Note that since there are 2π radians in a whole circle, that is why c = 2πr
For another angle θ, the arc length (distance around the circumference) is s = rθ