the hour hand of a clock is 5 inches long and the minute hand is 6 inches long. determine the distance between the tips of the hands at 11:40 pm????

at 11:40 pm the hour hand only has 1/36 of a complete revolution to go. That's 10 degrees

so, the tip of the hour hand is at

(5,pi/2+pi/18) in polar coordinates. That is
(-.868,4.924) in x-y coordinates

The tip of the minute hand is at
(6,-2pi/3), or
(-2.500,-4.330)

so, the tips of the hands are
√((-2.5+.868)^2 + (-4.33-4.924)^2)
= √(2.663 + 85.636)
= √88.3
= 9.4 in

Makes sense, since at 6:00 they'd be 11.0 inches apart.

To find the distance between the tips of the hour and minute hands at 11:40 pm, we need to calculate the position of each hand and then measure the distance between the two points.

First, let's determine the position of the hour hand. At 11:40 pm, the hour hand will be pointing between the 11 and 12 on the clock face. To find the exact position, we need to consider the fact that the hour hand moves 360 degrees in 12 hours, or 30 degrees per hour. At 11:40 pm, the hour hand has been moving for 11 hours and 40 minutes, which is approximately 11.67 hours.

So, the position of the hour hand can be calculated as follows:

Position of the hour hand = (Angle moved per hour) × (Number of hours) = 30 degrees/hour × 11.67 hours = 350.1 degrees.

Next, let's determine the position of the minute hand. At 11:40 pm, the minute hand will be pointing at the 8 on the clock face. Since the minute hand moves 360 degrees in 60 minutes, we can calculate its position using the following formula:

Position of the minute hand = (Angle moved per minute) × (Number of minutes) = 6 degrees/minute × 40 minutes = 240 degrees.

Now that we have the positions of both hands, we can calculate the distance between their tips. This can be done using basic trigonometry. The hands form a right-angled triangle, where the distance between their tips represents the hypotenuse.

Using the Pythagorean theorem, we have:

Distance = √(hour hand length^2 + minute hand length^2 - 2 × hour hand length × minute hand length × cos(angle between the hands)).

Plugging in the values:

Distance = √(5^2 + 6^2 - 2 × 5 × 6 × cos(angle between the hands)).

To find the angle between the hands, we subtract the position of the hour hand from the position of the minute hand:

Angle between the hands = Position of the minute hand - Position of the hour hand = 240 degrees - 350.1 degrees = -110.1 degrees.

Since cosine is a periodic function, we can ignore the negative sign and use the absolute value of the angle.

Plugging in the values:

Distance = √(5^2 + 6^2 - 2 × 5 × 6 × cos(abs(-110.1 degrees))).

Using a calculator to evaluate the cosine of the absolute angle, we get:

Distance ≈ 6.3 inches.

Therefore, the distance between the tips of the hands at 11:40 pm is approximately 6.3 inches.