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. 8:20

Accepted Answer

Hello, everyone. We are told the hands of a clock form two angles. We want to determine the smaller angle formed by the time 10, 35 our answer choices are a 96 degrees, 30 minutes. B 118 degrees, 30 minutes. C 107 degrees, 30 minutes. D 82 degrees, 30 minutes. So I'm gonna begin by thinking about my hour hand. So since we're talking about a clock, we're thinking about the degrees from the 12, not necessarily from standard position. So since a full rotation around our clock is 360 degrees, and we have 12 hours on a standard clock to find out how far the hour hand goes per hour. We're gonna do 360 degrees divided by 12, which is 30 degrees. So we know our, our hand goes 30 degrees per hour. Now, recall that the hour hand doesn't only move on the hour. It ticks along a little bit minute by minute. So we need to find out within the 30 degrees it goes per hour. How far does it go each minute? So we're gonna do 30 degrees divided by 60 minutes, which gives us 0.5 degrees, which is the same thing as 30 minutes. And that's how far the hour hand goes per minute. So now to find out at 10 35 we know we have 10 hours and minutes. So we'll see what the angle is from the 12 to the hour hand at 35. So we have 30 degrees multiplied by 10 for hours. Plus I'm gonna use 0.5 degrees multiplied by 35 for minutes. So we get 300 degrees plus 17.5 degrees. So at 10 35 our, our hand is 317. degrees from the 12th. Next, we're gonna work with the minute hand. So our minute hand makes a full rotation around the clock. So 360 degrees in one hour. So one hour is 60 minutes. So we're gonna do 360 degrees divided by 60 to find that the minute hand moves six degrees per minute. So we don't have to worry about the hour part here. We know that we are working with 35 minutes past the hour. So we're gonna do six degrees multiplied by minutes, which gives us 210 degrees. So at 10 35 the minute hand is 210 degrees from the 12, the hour hand is 317.5 degrees from the 12. So to find the smaller angle that is made here, we're gonna subtract them. So we're gonna do 317.5 degrees minus 210 degrees. And this will give us 107.5 degrees. And we don't want to leave this with a 0.5. So this will be 107 degrees and half a degree is the same as 30 minutes. So our angle formed here will be 107 degrees, 30 minutes, which is answer choice C have a nice day.

Find the measure of the smaller angle formed by the hands of a clock at the following times. 8:20

Key Concepts

Angle Calculation of Clock Hands

Converting Time to Angles

Determining the Smaller Angle

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