Questions: The minute hand of a clock is 7 inches long. How far does the tip of the minute hand move in 35 minutes? How far does it move in 40 minutes? The tip of the minute hand moves 25.66 inches in 35 minutes. (Type an integer or decimal rounded to two decimal places as needed.) The tip of the minute hand moves inches in 40 minutes. (Type an integer or decimal rounded to two decimal places as needed.)

Solution Steps

To find the distance the tip of the minute hand moves, we need to calculate the arc length it travels. The minute hand completes a full circle (360 degrees) in 60 minutes. We can use the proportion of the time given to find the fraction of the circle it travels and then use the formula for the arc length, which is \( \text{Arc Length} = \theta \times r \), where \( \theta \) is the angle in radians and \( r \) is the radius (length of the minute hand).

1. Convert the time to the fraction of the circle.
2. Convert the fraction to radians (since \( 2\pi \) radians is a full circle).
3. Use the arc length formula to find the distance.

The minute hand completes a full circle (360 degrees) in 60 minutes. To find the fraction of the circle traveled in a given number of minutes, we use the formula: \[ \text{Fraction of the circle} = \frac{\text{minutes}}{60} \]

For 35 minutes: \[ \text{Fraction of the circle} = \frac{35}{60} = \frac{7}{12} \]

For 40 minutes: \[ \text{Fraction of the circle} = \frac{40}{60} = \frac{2}{3} \]

Since \(2\pi\) radians is a full circle, we convert the fraction of the circle to radians: \[ \theta = \text{Fraction of the circle} \times 2\pi \]

For 35 minutes: \[ \theta_{35} = \frac{7}{12} \times 2\pi = \frac{7\pi}{6} \]

For 40 minutes: \[ \theta_{40} = \frac{2}{3} \times 2\pi = \frac{4\pi}{3} \]

The arc length \(s\) is given by the formula: \[ s = \theta \times r \] where \(r\) is the radius of the circle (length of the minute hand).

For 35 minutes: \[ s_{35} = \frac{7\pi}{6} \times 7 \approx 25.66 \text{ inches} \]

For 40 minutes: \[ s_{40} = \frac{4\pi}{3} \times 7 \approx 29.32 \text{ inches} \]

Final Answer