Find the measure of the smaller angle formed by the hands of a...

Question

Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

Accepted Answer

Hello, everyone. We are told the hands of the clock form two angles. We are asked to determine the smaller angle formed at the time. 5 25. Our answer choices are a 12 degrees, 30 minutes. B seven degrees 30 minutes C 15 degrees, 15 minutes. D nine degrees, 15 minutes. So first, I'm gonna think about my hour hand. I know that a full rotation around the clock is 360 degrees. So if there are 12 hours on the clock to find out how far the hand goes from the 12 every hour, we do 360 degrees divided by 12. And this tells us that we go 30 degrees per hour and that's how far away from 12 we go every hour. Now, unfortunately, the hour hand doesn't only move on the hour. It moves a little bit every minute. So to find out how far it goes every minute, we take our 30 degrees, we're gonna divide it by 60 minutes. So we'll find out how far does it go each single minute? And that would be 0.5 degrees. Or it could also be referred to as 30 minutes. Her minute. Of time. So to find the angle of our, our hand at 5 25 we have five hours and 25 minutes. So we're gonna use the two things we just found. So 30 degrees will be multiplied by the five hours that have already passed. Plus we're gonna use the decimal 0.5 degrees multiplied by the 25 minutes that have passed. So 30 degrees multiplied by five is 150 degrees 0.5 degrees multiplied by 25 is 12.5 degrees. So together this is 162.5 degrees. So we know the hour hand is 162.5 degrees away from the 12 at 5 25. Let's find out where the minute hand is. So thinking about a clock, I recall that the minute hand makes a full rotation every hour. So it goes all 360 degrees in one hour, which is 60 minutes. So we're gonna do 360 degrees divided by 60. So the minute hand goes six degrees per minute. This is enough for us to be able to figure out where the minute hand would be at 25. So it'd be 25 minutes past the hour. So 25 multiplied by six degrees is 150 degrees. So now that we know the location of both hands to find the smaller angle between them, we subtract. So we'll do 162.5 degrees minus 150 degrees, which gives us 12.5 degrees. We wanna put this in proper form. So we're gonna make it 12 degrees and 0.5 in minutes. So 0.5 is half, half of one degree is 30 minutes. So this smaller angle is 12 degrees, 30 minutes, which is answer choice a have a nice day.

Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

Key Concepts

Angle Measurement of Clock Hands

Calculating the Angle Between Two Lines

Time to Angle Conversion for the Hour Hand

Watch next