SOLUTION: The hour hand of a clock is half the length of the minute hand. How many more times area will the minute hand swept than the hour hand if they both move through the same angle?

Question 1118301: The hour hand of a clock is half the length of the minute hand. How many more times area will the minute hand swept than the hour hand if they both move through the same angle?
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
The hour hand of a clock is half the length of the minute hand.
How many more times area will the minute hand swept than the hour hand if they both move through the same angle?
:
let r = the radius of the hr hand
then
2r = radius of the minute hand
:
= =
Cancel
4 times
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The solution by the other tutor gets the answer by comparing the areas of the whole circles swept out by the hour and minute hands. Without saying so, the implication then is that the area swept out by the minute hand is 4 times the area swept out by the hour hand for any given angle.

But you don't need to use any area formulas for problems like this.

For any given angle, the regions swept out by the minute hand and the hour hand are similar figures -- equal fractions of a circle.

A general principle of geometry that makes many problems easy that look much more difficult is that, for similar figures with a scale factor (ratio of linear measurements) of a:b, the ratio of corresponding area measurements is a^2:b^2, and the ratio of corresponding volume measurements is a^3:b^3.

In this problem, you can immediately conclude that, since the length of the minute hand is 2 times the length of the minute hand, the area swept out by the minute hand is 2^2 = 4 times the area swept out by the minute hand.

So the answer to the problem is that the minute hand sweeps out 4 times as much area as the hour hand when they move through the same angle.

Note, however, that the exact question is

"How many more times area will the minute hand swept than the hour hand if they both move through the same angle?"

Correct language means that "4 times as much as" means "3 times more than"; so the real answer to the question that is asked is "3".

That is, the minute hand sweeps out 3 times more area then the hour hand when they move through the same angle; 3 times more than means 4 times as much, which is the correct answer.
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