# Math - Trig

posted by .

Hi I have a Trig question that i don't understand, can someone please explain how to solve for it using cosine law??

A Clock with a radius of 15 cm has an 11 cm minute hand and a 7 cm hour hand. How far apart, to the nearest centimetre, are the tips of the hands at each time?

a) 3:30 pm b) 6:38 am

• Math - Trig -

Consider a starting time of 12:00.
Every increment of time creates a new triangle with sides of 7 and 11 and an included angle of 5.4m which can be solved using the Law of Cosines.

Where did the 5.4 m come from?
The minute hand of a clock moves 360/60 = 6º/minute or 6º/m.
The hour hand of a clock moves 30/60 = .50º/m.
For example, at 25 minutes after 12:00, the minute hand has rotated 6(25) = 150º while the hour hand has rotated .50(25) = 12.5º. Therefore, the two hands are now 150 - 12.5 = 137.5º apart.
In general, the angle µ between the two hands is 5.5m up to the point when they are 180º apart.

We therefore, now have a triangle with sides of 7 and 11 and an included angle of 137.5º.

Using the Law of Cosines, a^2 = b^2 + c^2 - 2abcosA or a^2 = 7^2 + 11^2 - 2(7)11(cos137.5) = 16.83cm.

What happens beyond the point where the two hands are 180º apart you ask?

When 5.5m is greater than 180º, you subtract the result from 360º to derive the angle between the two hands.

Thus, at 12:40, µ = 360 - 5.5(40) = 140º.
At 1:00, the two hands are 30º apart.

After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.

You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - .5m = 180 or m = 32.7272 minutes after 1:00.

I'll let you continue from here.

• Math - Trig -

A slight correction to the last expression result in my previous response.

After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.

You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - (30 + .5m) = 180 or m = 38.1818 minutes after 1:00.

I'll let you continue from here.

• Math - Trig -

find in degrees and radians the angle between the hour hand and the minute hand of a clock at half past three

## Similar Questions

1. ### Math Trig

As the time changes from 4:10 pm to 6:45 pm on the analog clock, determine the change in radian measure of the minute hand.
2. ### Trig

Linear and angular speed Find w for each of the following: 1. the minute hand of a clock 2. the second hand of a clock The answer to #1 should be pi/30, but I do not understand how to solve these.
3. ### trig

The minute hand of a clock is 6 inches long. What distance does its tip move in 16 minutes?
4. ### trig

Through how many degrees does the minute hand of a clock moves in 5 minutes?
5. ### Math

The minute hand of a clock is 6 inches long and moves from 3 o clock to 5 o clock. how far does the tip of the minute hand move. Express your answer round to two decimal places. Can you solve this?
6. ### Cosine Law

The minute hand of a clock is 12 cm long, and the hour hand is 10cm long. Determine the distance between the tips of the hands at 9:30pm, to the nearest mm.
7. ### maths

the minute hand of a clock is 9cm long. the hour hand clock is 2/3 times longer than the minute clock. how much more distance will minute hand clock move than hour clock in 1 hour?
8. ### trig

The minute hand of a clock is 7 inches long and moves from 12 to 11 ​o'clock. How far does the tip of the minute hand​ move?
9. ### Math (Trig)

A clock tower has been constructed such that the center of the clock is 40 feet above ground. The minute hand on the clock is 3 feet long. How far above the ground the end of the minute hand at 12:05?
10. ### Algebra 2

Neela's clock is broken the minute hand rotates around the clock correctly but the hour hand is stuck in the three o'clock position. Suppose neela first looks at the clock when the hands are aligned and it shows 3:15. She looks at …

More Similar Questions