You look at yourself in a shiny 8.4-cm-diameter Christmas tree...

Question

You look at yourself in a shiny 8.4-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the ball’s front surface, where is your image? Is it real or virtual? Is it upright or inverted?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A car has a convex side mirror with an effective diameter of about 12 centimeters. Where do you see your reflection if you look into this mirror from 40 centimeters away? Is it real or virtual? Is it right side up or upside down? Awesome. So it appears for this particular problem we're asked to solve for three separate answers. So we're trying to figure out from the all the information that's provided by the problem itself. We're trying to figure out where we see our reflection in this car's convex mirror. That's our first answer. We're trying to determine whether or not this image is real or virtual. And we're also asked to figure out if it's right side up or upside down. And it also appears when we kind of briefly look over our multiple choice answers, which we read those off together in just a second. But we're also trying to figure out if it's located. Actually, it's already straight, straight up tells us that it's located behind its surface. So behind the surface of the mirror is what it states in the promise. So let's read off our multiple choice answers to see what our final answer set might be. So A is the reflection will be 2.8 centimeters from the mirror virtual upright and located behind the its surface B is, the reflection will be 2.8 centimeters from the mirror real upright. And located behind its surface C is the reflection will be 3.8 centimeters from the mirror real inverted and located behind its surface. And finally, D is the reflection will be 3.8 centimeters from the mirror virtual inverted and located behind its surface. OK. So first off, we need to calculate the focal length. But before we do that, let's quickly write down all of our known variables. So we know the diameter of the car side mirror and let's call this diameter D and the diameter D is equal to 12 centimeters. And we also know that the radius of curvature of the side mirror, let's call this RC radius of curvature is equal to six centimeters. Awesome. So with that in mind, we need to recall that for the mirror, the focal length F is half the radius of curvature and is negative. Therefore, as we should recall, we should write our equation for the focal length as the focal length F is equal to negative R which R is the radius divided by two or negative R divided by two. So when we plug in our known variables, we will find that F the focal length is equal to negative six divided by two, which will mean that F the focal length is equal to. And don't forget that our units are in centimeters. Never forget to write your units. So that would mean when we plug it into our calculator, our final answer is negative three centimeters. Awesome. So now we need to recall and use the mirror equation which states that one divided by F where F is the focal length is equal to one divided by. So where, so is the object distance plus one divided by SI where si is the image distance? Awesome. So let's quickly note really quick that we know the value for. So we know that the object distance is equal to 40 centimeters. However, however, we do not know what SI is. We do not know what the image distance is. And because we don't know what the image distance is, however, we know what the value for the focal length is and we know what the value for the object distance is. So let's plug in all of our known variables. And let's also rearrange this equation to isolate and sulfur si all by itself. So when we do that, we will find that one divided by si is equal to one divided by F minus one divided by. So and now we need to use some algebra to rearrange this equation to isolate sulfur SS I. But let's do it in pieces here to make it more simple. So let's plug in our known variables. So we know that one divided by si is equal to one divided by negative three centimeters minus one divided by 40 centimeters. Awesome, which is all equal to negative 0.358. And don't forget that a tuts are centimeters to the power of negative one. Awesome. So now in order to get si completely by itself, we need to take the inverse. So si is equal to one divided by negative 0.358 centimeters. Awesome. So let me plug that into our calculator and we round to approximately two significant figures. We will find that SI is approximately equal to negative 2.8 centimeters. And that is our final answer for the reflection. Awesome. We did it. So the negative value indicates that the image is virtual and it's because our that is our value here for SI is negative. So the image is virtual and it will be behind the surface of the side mirror. So behind Awesome. So now we need to solve for one last detail here. So we need to note that our final step is we need to recall and use our magnification equation which states that the magnification M is equal to negative SI divided by. So awesome. So when we plug in our known variables, when we solve for M, you will find that M is equal to negative negative 2.79. Because we're gonna use to be more precise. We're gonna use our non rounded number that we got for si. So let me make a quick little note of this, I'll write it in red. So initially when you type this into your calculator, you'll get negative two 0.79 centimeters. And when we round to one decimal place or to two significant figures, that's how we get negative 2.8. But in order to keep our answer precise, let's use our unrolled number for our magnification equation. So negative 2.79 centimeters divided by 40 centimeters, which is our object distance. So 40 centimeters. So when we plug that into our calculator, we will get positive 0.06975 centimeters. So when we round and that's when we round to 123455 decimal places. So this means without a doubt that this proves that because our answer is a positive value. Since we have a positive value for the magnification, this will indicate that the image is upright. Fantastic. So once again, this is important to write down. So if the magnification value is positive, that will indicate that the image is upright rather if it was a negative value for the magnification, that would mean that the image is upside down? Awesome. So with that in mind, we can, we can say with certainty, the image is virtual upright and located behind the surface of the mirror. And this means that our final answer has to be the letter A which states that the reflection will be 2.8 centimeters from the mirror virtual upright and located behind its surface. Thank you so much for watching. Hopefully, that helped and I can't wait to see in the next video. Bye.

You look at yourself in a shiny 8.4-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the ball’s front surface, where is your image? Is it real or virtual? Is it upright or inverted?

Key Concepts

Concave Mirrors

Image Formation

Mirror Equation and Magnification

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