The minute hand and the hour hand of a clock have lengths

inches and inches, respectively. Determine the distance
between the tips of the hands at 10:00 in terms of and m h.

To determine the distance between the tips of the hour and minute hands at 10:00, we need to use some basic geometry and trigonometry.

First, let's find the angles that the hour and minute hands make with the 12 o'clock position.

At 10:00, the minute hand points directly at the 12, so its angle with the 12 o'clock position is 0°.

The hour hand, on the other hand, is halfway between the 10 and 11, which is equivalent to 10 + 1/2 = 10.5 hours. Since there are 12 hours on a clock, this is θ = (10.5 / 12) * 360° = 315°.

Next, we can use the cosine law to find the distance between the tips of the hands.

The distance between the tips of the hands is the hypotenuse of a right triangle. The hour hand length, x inches, represents one side of the triangle, and the minute hand length, y inches, represents the other side. The angle between the sides is the difference between the angles made by the hour and minute hands with the 12 o'clock position.

Using the cosine law, we have:

c² = x² + y² - 2xy * cos(θ)

where c is the distance between the tips of the hands and θ is the angle between the hour and minute hands.

Plugging in the values, we get:

c² = x² + y² - 2xy * cos(315°)

Now we can find the distance between the tips of the hands by taking the square root of c²:

c = sqrt(x² + y² - 2xy * cos(315°))

So, the distance between the tips of the hands at 10:00 is sqrt(x² + y² - 2xy * cos(315°)).

At 10:00 the hands are 60 degrees apart. So, as soon as you decide how long the hands are, use the law of cosines to find the desired distance.