In Exercises 87–90, find the absolute value of the radian measure...

Question

In Exercises 87–90, find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time.

3 minutes and 40 seconds

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the angle and radiance at which the second hand of a clock rotates through the following time. Consider the clockwise rotation is positive. The time is five minutes and 12 seconds. Our answer choices are answer choice. A 52 pi divided by five answer choice. B 65 pi divided by six answer choice. C 117 pi divided by 10 and answer choice. D 74 pi divided by 15. All right. So what have we got here? Well, we're talking about the second hand of a clock. So we're not talking about standard position of our angle. We've got a clock here. And as it says, the clockwise rotation here is positive. Well, where is our initial side for a clock that's pointing from right from the center. It's pointing straight up at 12 o'clock. There's our initial side and we're talking about a second hand and as a second hand rotates as it completes one full revolution. How many seconds is that? Well, it takes 60 seconds for one complete revolution. So we want to determine the angle after five minutes and 12 seconds. We want to know for our terminal side, where is the second hand, after five minutes and 12 seconds? We don't know. So, how do we figure that out and noting that it needs to be figured out in radiance. Well, how many seconds has it been going for so far? We know it's five minutes and 12 seconds. Well, how many seconds is five minutes? We take five minutes. We know that there are 60 seconds in one minute. So we can multiply five minutes by seconds, divided by one minute. Then we have our minutes canceling out and we take five times 60 which is 300. So we have 300 seconds. This has been going for plus the 12 seconds that we know about this second hand has been going for 312 seconds. And now to determine where it is after 312 seconds, we can set up a ratio so we can take our 312 seconds that we know about and divide that by what's our base here? Well, one full rotation, as we said is 60 seconds. So the first fraction for our ratio is 312 divided by that's equal to. Now, we need to convert this to radiance. So what's our base for radiance? Well, we recall from previous lessons, that one full rotation around a circle is two pi when we're talking about radiance. So the base of our fraction on the right hand side here is two pi that's our denominator. What's our numerator? Well, that's what we're trying to find. So we can label that as X now to solve. For X. Here, we can multiply both sides by two pi. So taking two pi multiplied by 312 in the numerator and dividing it by 60 when we do that and simplify, it will be 52 pi divided by five. And that is our angle in radiance. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 87–90, find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 3 minutes and 40 seconds

Key Concepts

Radian Measure

Angular Velocity

Time Conversion

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